Find the Maximum/Minimum Value f(x)=x^4-4x^3+7

$f (x) = x^{4} - 4 x^{3} + 7$

Step 1

Find the first derivative of the function.

Differentiate.

By the Sum Rule, the derivative of $x^{4} - 4 x^{3} + 7$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [7]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [7]$

Evaluate $\frac{d}{d x} [- 4 x^{3}]$ .

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{3}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{3}]$ .

$4 x^{3} - 4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [7]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 x^{3} - 4 (3 x^{2}) + \frac{d}{d x} [7]$

Multiply $3$ by $- 4$ .

$4 x^{3} - 12 x^{2} + \frac{d}{d x} [7]$

Differentiate using the Constant Rule.

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$4 x^{3} - 12 x^{2} + 0$

Add $4 x^{3} - 12 x^{2}$ and $0$ .

$4 x^{3} - 12 x^{2}$

Step 2

Find the second derivative of the function.

By the Sum Rule, the derivative of $4 x^{3} - 12 x^{2}$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 12 x^{2})$

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 12 x^{2})$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 12 x^{2})$

Multiply $3$ by $4$ .

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

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

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

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

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

Multiply $2$ by $- 12$ .

$f'' (x) = 12 x^{2} - 24 x$

Step 3

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

$4 x^{3} - 12 x^{2} = 0$

Step 4

Find the first derivative.

Differentiate.

By the Sum Rule, the derivative of $x^{4} - 4 x^{3} + 7$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [7]$ .

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 4 x^{3}) + \frac{d}{d x} (7)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f' (x) = 4 x^{3} + \frac{d}{d x} (- 4 x^{3}) + \frac{d}{d x} (7)$

Evaluate $\frac{d}{d x} [- 4 x^{3}]$ .

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{3}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 4 x^{3} - 4 \frac{d}{d x} x^{3} + \frac{d}{d x} (7)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = 4 x^{3} - 4 (3 x^{2}) + \frac{d}{d x} (7)$

Multiply $3$ by $- 4$ .

$f' (x) = 4 x^{3} - 12 x^{2} + \frac{d}{d x} (7)$

Differentiate using the Constant Rule.

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$f' (x) = 4 x^{3} - 12 x^{2} + 0$

Add $4 x^{3} - 12 x^{2}$ and $0$ .

$f' (x) = 4 x^{3} - 12 x^{2}$

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 12 x^{2}$ .

$4 x^{3} - 12 x^{2}$

Step 5

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 12 x^{2} = 0$ .

Set the first derivative equal to $0$ .

$4 x^{3} - 12 x^{2} = 0$

Factor $4 x^{2}$ out of $4 x^{3} - 12 x^{2}$ .

Factor $4 x^{2}$ out of $4 x^{3}$ .

$4 x^{2} (x) - 12 x^{2} = 0$

Factor $4 x^{2}$ out of $- 12 x^{2}$ .

$4 x^{2} (x) + 4 x^{2} (- 3) = 0$

Factor $4 x^{2}$ out of $4 x^{2} (x) + 4 x^{2} (- 3)$ .

$4 x^{2} (x - 3) = 0$

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

$x^{2} = 0$

$x - 3 = 0$

Set $x^{2}$ equal to $0$ and solve for $x$ .

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Solve $x^{2} = 0$ for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $4 x^{2} (x - 3) = 0$ true.

$x = 0, 3$

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.

$x = 0, 3$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 {(0)}^{2} - 24 \cdot 0$

Step 9

Evaluate the second derivative.

Simplify each term.

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

$12 \cdot 0 - 24 \cdot 0$

Multiply $12$ by $0$ .

$0 - 24 \cdot 0$

Multiply $- 24$ by $0$ .

$0 + 0$

Add $0$ and $0$ .

$0$

Step 10

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, 0) \cup (0, 3) \cup (3, \infty)$

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $4 x^{3} - 12 x^{2}$ to check if the result is negative or positive.

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = 4 {(- 2)}^{3} - 12 {(- 2)}^{2}$

Simplify the result.

Simplify each term.

Raise $- 2$ to the power of $3$ .

$f' (- 2) = 4 \cdot - 8 - 12 {(- 2)}^{2}$

Multiply $4$ by $- 8$ .

$f' (- 2) = - 32 - 12 {(- 2)}^{2}$

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

$f' (- 2) = - 32 - 12 \cdot 4$

Multiply $- 12$ by $4$ .

$f' (- 2) = - 32 - 48$

Subtract $48$ from $- 32$ .

$f' (- 2) = - 80$

The final answer is $- 80$ .

$- 80$

Substitute any number, such as $2$ , from the interval $(0, 3)$ in the first derivative $4 x^{3} - 12 x^{2}$ to check if the result is negative or positive.

Replace the variable $x$ with $2$ in the expression.

$f' (2) = 4 {(2)}^{3} - 12 {(2)}^{2}$

Simplify the result.

Simplify each term.

Raise $2$ to the power of $3$ .

$f' (2) = 4 \cdot 8 - 12 {(2)}^{2}$

Multiply $4$ by $8$ .

$f' (2) = 32 - 12 {(2)}^{2}$

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

$f' (2) = 32 - 12 \cdot 4$

Multiply $- 12$ by $4$ .

$f' (2) = 32 - 48$

Subtract $48$ from $32$ .

$f' (2) = - 16$

The final answer is $- 16$ .

$- 16$

Substitute any number, such as $6$ , from the interval $(3, \infty)$ in the first derivative $4 x^{3} - 12 x^{2}$ to check if the result is negative or positive.

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

$f' (6) = 4 {(6)}^{3} - 12 {(6)}^{2}$

Simplify the result.

Simplify each term.

Raise $6$ to the power of $3$ .

$f' (6) = 4 \cdot 216 - 12 {(6)}^{2}$

Multiply $4$ by $216$ .

$f' (6) = 864 - 12 {(6)}^{2}$

Raise $6$ to the power of $2$ .

$f' (6) = 864 - 12 \cdot 36$

Multiply $- 12$ by $36$ .

$f' (6) = 864 - 432$

Subtract $432$ from $864$ .

$f' (6) = 432$

The final answer is $432$ .

$432$

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Since the first derivative changed signs from negative to positive around $x = 3$ , then $x = 3$ is a local minimum.

$x = 3$ is a local minimum

Step 11

Find the Maximum/Minimum Value f(x)=x^4-4x^3+7

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