Find the Maximum/Minimum Value x(36-2x)^2

$x {(36 - 2 x)}^{2}$

Step 1

Find the first derivative of the function.

Rewrite ${(36 - 2 x)}^{2}$ as $(36 - 2 x) (36 - 2 x)$ .

$\frac{d}{d x} [x ((36 - 2 x) (36 - 2 x))]$

Expand $(36 - 2 x) (36 - 2 x)$ using the FOIL Method.

Apply the distributive property.

$\frac{d}{d x} [x (36 (36 - 2 x) - 2 x (36 - 2 x))]$

Apply the distributive property.

$\frac{d}{d x} [x (36 \cdot 36 + 36 (- 2 x) - 2 x (36 - 2 x))]$

Apply the distributive property.

$\frac{d}{d x} [x (36 \cdot 36 + 36 (- 2 x) - 2 x \cdot 36 - 2 x (- 2 x))]$

Simplify and combine like terms.

Simplify each term.

Multiply $36$ by $36$ .

$\frac{d}{d x} [x (1296 + 36 (- 2 x) - 2 x \cdot 36 - 2 x (- 2 x))]$

Multiply $- 2$ by $36$ .

$\frac{d}{d x} [x (1296 - 72 x - 2 x \cdot 36 - 2 x (- 2 x))]$

Multiply $36$ by $- 2$ .

$\frac{d}{d x} [x (1296 - 72 x - 72 x - 2 x (- 2 x))]$

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [x (1296 - 72 x - 72 x - 2 \cdot - 2 x \cdot x)]$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{d}{d x} [x (1296 - 72 x - 72 x - 2 \cdot - 2 (x \cdot x))]$

Multiply $x$ by $x$ .

$\frac{d}{d x} [x (1296 - 72 x - 72 x - 2 \cdot - 2 x^{2})]$

Multiply $- 2$ by $- 2$ .

$\frac{d}{d x} [x (1296 - 72 x - 72 x + 4 x^{2})]$

Subtract $72 x$ from $- 72 x$ .

$\frac{d}{d x} [x (1296 - 144 x + 4 x^{2})]$

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 1296 - 144 x + 4 x^{2}$ .

$x \frac{d}{d x} [1296 - 144 x + 4 x^{2}] + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} [x]$

Differentiate.

By the Sum Rule, the derivative of $1296 - 144 x + 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1296] + \frac{d}{d x} [- 144 x] + \frac{d}{d x} [4 x^{2}]$ .

$x (\frac{d}{d x} [1296] + \frac{d}{d x} [- 144 x] + \frac{d}{d x} [4 x^{2}]) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} [x]$

Since $1296$ is constant with respect to $x$ , the derivative of $1296$ with respect to $x$ is $0$ .

$x (0 + \frac{d}{d x} [- 144 x] + \frac{d}{d x} [4 x^{2}]) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} [x]$

Add $0$ and $\frac{d}{d x} [- 144 x]$ .

$x (\frac{d}{d x} [- 144 x] + \frac{d}{d x} [4 x^{2}]) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} [x]$

Since $- 144$ is constant with respect to $x$ , the derivative of $- 144 x$ with respect to $x$ is $- 144 \frac{d}{d x} [x]$ .

$x (- 144 \frac{d}{d x} [x] + \frac{d}{d x} [4 x^{2}]) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} [x]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x (- 144 \cdot 1 + \frac{d}{d x} [4 x^{2}]) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} [x]$

Multiply $- 144$ by $1$ .

$x (- 144 + \frac{d}{d x} [4 x^{2}]) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} [x]$

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$x (- 144 + 4 \frac{d}{d x} [x^{2}]) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} [x]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x (- 144 + 4 (2 x)) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} [x]$

Multiply $2$ by $4$ .

$x (- 144 + 8 x) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} [x]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x (- 144 + 8 x) + (1296 - 144 x + 4 x^{2}) \cdot 1$

Multiply $1296 - 144 x + 4 x^{2}$ by $1$ .

$x (- 144 + 8 x) + 1296 - 144 x + 4 x^{2}$

Simplify.

Apply the distributive property.

$x \cdot - 144 + x (8 x) + 1296 - 144 x + 4 x^{2}$

Combine terms.

Move $- 144$ to the left of $x$ .

$- 144 \cdot x + x (8 x) + 1296 - 144 x + 4 x^{2}$

Raise $x$ to the power of $1$ .

$- 144 x + 8 (x^{1} x) + 1296 - 144 x + 4 x^{2}$

Raise $x$ to the power of $1$ .

$- 144 x + 8 (x^{1} x^{1}) + 1296 - 144 x + 4 x^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 144 x + 8 x^{1 + 1} + 1296 - 144 x + 4 x^{2}$

Add $1$ and $1$ .

$- 144 x + 8 x^{2} + 1296 - 144 x + 4 x^{2}$

Subtract $144 x$ from $- 144 x$ .

$- 288 x + 8 x^{2} + 1296 + 4 x^{2}$

Add $8 x^{2}$ and $4 x^{2}$ .

$- 288 x + 12 x^{2} + 1296$

Reorder terms.

$12 x^{2} - 288 x + 1296$

Step 2

Find the second derivative of the function.

By the Sum Rule, the derivative of $12 x^{2} - 288 x + 1296$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 288 x] + \frac{d}{d x} [1296]$ .

$f'' (x) = \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (- 288 x) + \frac{d}{d x} (1296)$

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 288 x) + \frac{d}{d x} (1296)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = 12 (2 x) + \frac{d}{d x} (- 288 x) + \frac{d}{d x} (1296)$

Multiply $2$ by $12$ .

$f'' (x) = 24 x + \frac{d}{d x} (- 288 x) + \frac{d}{d x} (1296)$

Evaluate $\frac{d}{d x} [- 288 x]$ .

Since $- 288$ is constant with respect to $x$ , the derivative of $- 288 x$ with respect to $x$ is $- 288 \frac{d}{d x} [x]$ .

$f'' (x) = 24 x - 288 \frac{d}{d x} x + \frac{d}{d x} (1296)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = 24 x - 288 \cdot 1 + \frac{d}{d x} (1296)$

Multiply $- 288$ by $1$ .

$f'' (x) = 24 x - 288 + \frac{d}{d x} (1296)$

Differentiate using the Constant Rule.

Since $1296$ is constant with respect to $x$ , the derivative of $1296$ with respect to $x$ is $0$ .

$f'' (x) = 24 x - 288 + 0$

Add $24 x - 288$ and $0$ .

$f'' (x) = 24 x - 288$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$12 x^{2} - 288 x + 1296 = 0$

Step 4

Find the first derivative.

Rewrite ${(36 - 2 x)}^{2}$ as $(36 - 2 x) (36 - 2 x)$ .

$f' (x) = \frac{d}{d x} (x ((36 - 2 x) (36 - 2 x)))$

Expand $(36 - 2 x) (36 - 2 x)$ using the FOIL Method.

Apply the distributive property.

$f' (x) = \frac{d}{d x} (x (36 (36 - 2 x) - 2 x (36 - 2 x)))$

Apply the distributive property.

$f' (x) = \frac{d}{d x} (x (36 \cdot 36 + 36 (- 2 x) - 2 x (36 - 2 x)))$

Apply the distributive property.

$f' (x) = \frac{d}{d x} (x (36 \cdot 36 + 36 (- 2 x) - 2 x \cdot 36 - 2 x (- 2 x)))$

Simplify and combine like terms.

Simplify each term.

Multiply $36$ by $36$ .

$f' (x) = \frac{d}{d x} (x (1296 + 36 (- 2 x) - 2 x \cdot 36 - 2 x (- 2 x)))$

Multiply $- 2$ by $36$ .

$f' (x) = \frac{d}{d x} (x (1296 - 72 x - 2 x \cdot 36 - 2 x (- 2 x)))$

Multiply $36$ by $- 2$ .

$f' (x) = \frac{d}{d x} (x (1296 - 72 x - 72 x - 2 x (- 2 x)))$

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{d}{d x} (x (1296 - 72 x - 72 x - 2 \cdot (- 2 x \cdot x)))$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$f' (x) = \frac{d}{d x} (x (1296 - 72 x - 72 x - 2 \cdot (- 2 (x \cdot x))))$

Multiply $x$ by $x$ .

$f' (x) = \frac{d}{d x} (x (1296 - 72 x - 72 x - 2 \cdot (- 2 x^{2})))$

Multiply $- 2$ by $- 2$ .

$f' (x) = \frac{d}{d x} (x (1296 - 72 x - 72 x + 4 x^{2}))$

Subtract $72 x$ from $- 72 x$ .

$f' (x) = \frac{d}{d x} (x (1296 - 144 x + 4 x^{2}))$

$f' (x) = x \frac{d}{d x} (1296 - 144 x + 4 x^{2}) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} (x)$

Differentiate.

By the Sum Rule, the derivative of $1296 - 144 x + 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1296] + \frac{d}{d x} [- 144 x] + \frac{d}{d x} [4 x^{2}]$ .

$f' (x) = x (\frac{d}{d x} (1296) + \frac{d}{d x} (- 144 x) + \frac{d}{d x} (4 x^{2})) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} (x)$

Since $1296$ is constant with respect to $x$ , the derivative of $1296$ with respect to $x$ is $0$ .

$f' (x) = x (0 + \frac{d}{d x} (- 144 x) + \frac{d}{d x} (4 x^{2})) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} (x)$

Add $0$ and $\frac{d}{d x} [- 144 x]$ .

$f' (x) = x (\frac{d}{d x} (- 144 x) + \frac{d}{d x} (4 x^{2})) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} (x)$

Since $- 144$ is constant with respect to $x$ , the derivative of $- 144 x$ with respect to $x$ is $- 144 \frac{d}{d x} [x]$ .

$f' (x) = x (- 144 \frac{d}{d x} x + \frac{d}{d x} (4 x^{2})) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} (x)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = x (- 144 \cdot 1 + \frac{d}{d x} (4 x^{2})) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} (x)$

Multiply $- 144$ by $1$ .

$f' (x) = x (- 144 + \frac{d}{d x} (4 x^{2})) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} (x)$

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$f' (x) = x (- 144 + 4 \frac{d}{d x} (x^{2})) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} (x)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = x (- 144 + 4 (2 x)) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} (x)$

Multiply $2$ by $4$ .

$f' (x) = x (- 144 + 8 x) + (1296 - 144 x + 4 x^{2}) \frac{d}{d x} (x)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = x (- 144 + 8 x) + (1296 - 144 x + 4 x^{2}) \cdot 1$

Multiply $1296 - 144 x + 4 x^{2}$ by $1$ .

$f' (x) = x (- 144 + 8 x) + 1296 - 144 x + 4 x^{2}$

Simplify.

Apply the distributive property.

$f' (x) = x \cdot - 144 + x (8 x) + 1296 - 144 x + 4 x^{2}$

Combine terms.

Move $- 144$ to the left of $x$ .

$f' (x) = - 144 \cdot x + x (8 x) + 1296 - 144 x + 4 x^{2}$

Raise $x$ to the power of $1$ .

$f' (x) = - 144 x + 8 (x \cdot x) + 1296 - 144 x + 4 x^{2}$

Raise $x$ to the power of $1$ .

$f' (x) = - 144 x + 8 (x \cdot x) + 1296 - 144 x + 4 x^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = - 144 x + 8 x^{1 + 1} + 1296 - 144 x + 4 x^{2}$

Add $1$ and $1$ .

$f' (x) = - 144 x + 8 x^{2} + 1296 - 144 x + 4 x^{2}$

Subtract $144 x$ from $- 144 x$ .

$f' (x) = - 288 x + 8 x^{2} + 1296 + 4 x^{2}$

Add $8 x^{2}$ and $4 x^{2}$ .

$f' (x) = - 288 x + 12 x^{2} + 1296$

Reorder terms.

$f' (x) = 12 x^{2} - 288 x + 1296$

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 288 x + 1296$ .

$12 x^{2} - 288 x + 1296$

Step 5

Set the first derivative equal to $0$ then solve the equation $12 x^{2} - 288 x + 1296 = 0$ .

Set the first derivative equal to $0$ .

$12 x^{2} - 288 x + 1296 = 0$

Factor the left side of the equation.

Factor $12$ out of $12 x^{2} - 288 x + 1296$ .

Factor $12$ out of $12 x^{2}$ .

$12 (x^{2}) - 288 x + 1296 = 0$

Factor $12$ out of $- 288 x$ .

$12 (x^{2}) + 12 (- 24 x) + 1296 = 0$

Factor $12$ out of $1296$ .

$12 x^{2} + 12 (- 24 x) + 12 \cdot 108 = 0$

Factor $12$ out of $12 x^{2} + 12 (- 24 x)$ .

$12 (x^{2} - 24 x) + 12 \cdot 108 = 0$

Factor $12$ out of $12 (x^{2} - 24 x) + 12 \cdot 108$ .

$12 (x^{2} - 24 x + 108) = 0$

Factor.

Factor $x^{2} - 24 x + 108$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $108$ and whose sum is $- 24$ .

$- 18, - 6$

Write the factored form using these integers.

$12 ((x - 18) (x - 6)) = 0$

Remove unnecessary parentheses.

$12 (x - 18) (x - 6) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 18 = 0$

$x - 6 = 0$

Set $x - 18$ equal to $0$ and solve for $x$ .

Set $x - 18$ equal to $0$ .

$x - 18 = 0$

Add $18$ to both sides of the equation.

$x = 18$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

The final solution is all the values that make $12 (x - 18) (x - 6) = 0$ true.

$x = 18, 6$

Step 6

Find the values where the derivative is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 18, 6$

Step 8

Evaluate the second derivative at $x = 18$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$24 (18) - 288$

Step 9

Evaluate the second derivative.

Multiply $24$ by $18$ .

$432 - 288$

Subtract $288$ from $432$ .

$144$

Step 10

$x = 18$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 18$ is a local minimum

Step 11

Find the y-value when $x = 18$ .

Replace the variable $x$ with $18$ in the expression.

$f (18) = (18) {(36 - 2 \cdot 18)}^{2}$

Simplify the result.

Multiply $- 2$ by $18$ .

$f (18) = 18 {(36 - 36)}^{2}$

Subtract $36$ from $36$ .

$f (18) = 18 \cdot 0^{2}$

Raising $0$ to any positive power yields $0$ .

$f (18) = 18 \cdot 0$

Multiply $18$ by $0$ .

$f (18) = 0$

The final answer is $0$ .

$y = 0$

Step 12

Evaluate the second derivative at $x = 6$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$24 (6) - 288$

Step 13

Evaluate the second derivative.

Multiply $24$ by $6$ .

$144 - 288$

Subtract $288$ from $144$ .

$- 144$

Step 14

$x = 6$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 6$ is a local maximum

Step 15

Find the y-value when $x = 6$ .

Replace the variable $x$ with $6$ in the expression.

$f (6) = (6) {(36 - 2 \cdot 6)}^{2}$

Simplify the result.

Multiply $- 2$ by $6$ .

$f (6) = 6 {(36 - 12)}^{2}$

Subtract $12$ from $36$ .

$f (6) = 6 \cdot 24^{2}$

Raise $24$ to the power of $2$ .

$f (6) = 6 \cdot 576$

Multiply $6$ by $576$ .

$f (6) = 3456$

The final answer is $3456$ .

$y = 3456$

Step 16

These are the local extrema for $f (x) = x {(36 - 2 x)}^{2}$ .

$(18, 0)$ is a local minima

$(6, 3456)$ is a local maxima

Step 17

Find the Maximum/Minimum Value x(36-2x)^2

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