Find the Concavity f(x)=9x^4-54x^2

$f (x) = 9 x^{4} - 54 x^{2}$

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

Find the second derivative.

Find the first derivative.

By the Sum Rule, the derivative of $9 x^{4} - 54 x^{2}$ with respect to $x$ is $\frac{d}{d x} [9 x^{4}] + \frac{d}{d x} [- 54 x^{2}]$ .

$f' (x) = \frac{d}{d x} (9 x^{4}) + \frac{d}{d x} (- 54 x^{2})$

Evaluate $\frac{d}{d x} [9 x^{4}]$ .

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{4}$ with respect to $x$ is $9 \frac{d}{d x} [x^{4}]$ .

$f' (x) = 9 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 54 x^{2})$

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

$f' (x) = 9 (4 x^{3}) + \frac{d}{d x} (- 54 x^{2})$

Multiply $4$ by $9$ .

$f' (x) = 36 x^{3} + \frac{d}{d x} (- 54 x^{2})$

Evaluate $\frac{d}{d x} [- 54 x^{2}]$ .

Since $- 54$ is constant with respect to $x$ , the derivative of $- 54 x^{2}$ with respect to $x$ is $- 54 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 36 x^{3} - 54 \frac{d}{d x} x^{2}$

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

$f' (x) = 36 x^{3} - 54 (2 x)$

Multiply $2$ by $- 54$ .

$f' (x) = 36 x^{3} - 108 x$

Find the second derivative.

By the Sum Rule, the derivative of $36 x^{3} - 108 x$ with respect to $x$ is $\frac{d}{d x} [36 x^{3}] + \frac{d}{d x} [- 108 x]$ .

$f'' (x) = \frac{d}{d x} (36 x^{3}) + \frac{d}{d x} (- 108 x)$

Evaluate $\frac{d}{d x} [36 x^{3}]$ .

Since $36$ is constant with respect to $x$ , the derivative of $36 x^{3}$ with respect to $x$ is $36 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 36 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 108 x)$

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

$f'' (x) = 36 (3 x^{2}) + \frac{d}{d x} (- 108 x)$

Multiply $3$ by $36$ .

$f'' (x) = 108 x^{2} + \frac{d}{d x} (- 108 x)$

Evaluate $\frac{d}{d x} [- 108 x]$ .

Since $- 108$ is constant with respect to $x$ , the derivative of $- 108 x$ with respect to $x$ is $- 108 \frac{d}{d x} [x]$ .

$f'' (x) = 108 x^{2} - 108 \frac{d}{d x} x$

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

$f'' (x) = 108 x^{2} - 108 \cdot 1$

Multiply $- 108$ by $1$ .

$f'' (x) = 108 x^{2} - 108$

The second derivative of $f (x)$ with respect to $x$ is $108 x^{2} - 108$ .

$108 x^{2} - 108$

Set the second derivative equal to $0$ then solve the equation $108 x^{2} - 108 = 0$ .

Set the second derivative equal to $0$ .

$108 x^{2} - 108 = 0$

Add $108$ to both sides of the equation.

$108 x^{2} = 108$

Divide each term in $108 x^{2} = 108$ by $108$ and simplify.

Divide each term in $108 x^{2} = 108$ by $108$ .

$\frac{108 x^{2}}{108} = \frac{108}{108}$

Simplify the left side.

Cancel the common factor of $108$ .

Cancel the common factor.

$\frac{108 x^{2}}{108} = \frac{108}{108}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{108}{108}$

Simplify the right side.

Divide $108$ by $108$ .

$x^{2} = 1$

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

$x = \pm \sqrt{1}$

Any root of $1$ is $1$ .

$x = \pm 1$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Step 2

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - 1) \cup (- 1, 1) \cup (1, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - 1)$ into the second derivative and evaluate to determine the concavity.

Replace the variable $x$ with $- 4$ in the expression.

$f'' (- 4) = 108 {(- 4)}^{2} - 108$

Simplify the result.

Simplify each term.

Raise $- 4$ to the power of $2$ .

$f'' (- 4) = 108 \cdot 16 - 108$

Multiply $108$ by $16$ .

$f'' (- 4) = 1728 - 108$

Subtract $108$ from $1728$ .

$f'' (- 4) = 1620$

The final answer is $1620$ .

$1620$

The graph is concave up on the interval $(- \infty, - 1)$ because $f'' (- 4)$ is positive.

Concave up on $(- \infty, - 1)$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(- 1, 1)$ into the second derivative and evaluate to determine the concavity.

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = 108 {(0)}^{2} - 108$

Simplify the result.

Simplify each term.

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

$f'' (0) = 108 \cdot 0 - 108$

Multiply $108$ by $0$ .

$f'' (0) = 0 - 108$

Subtract $108$ from $0$ .

$f'' (0) = - 108$

The final answer is $- 108$ .

$- 108$

The graph is concave down on the interval $(- 1, 1)$ because $f'' (0)$ is negative.

Concave down on $(- 1, 1)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(1, \infty)$ into the second derivative and evaluate to determine the concavity.

Replace the variable $x$ with $4$ in the expression.

$f'' (4) = 108 {(4)}^{2} - 108$

Simplify the result.

Simplify each term.

Raise $4$ to the power of $2$ .

$f'' (4) = 108 \cdot 16 - 108$

Multiply $108$ by $16$ .

$f'' (4) = 1728 - 108$

Subtract $108$ from $1728$ .

$f'' (4) = 1620$

The final answer is $1620$ .

$1620$

The graph is concave up on the interval $(1, \infty)$ because $f'' (4)$ is positive.

Concave up on $(1, \infty)$ since $f'' (x)$ is positive

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, - 1)$ since $f'' (x)$ is positive

Concave down on $(- 1, 1)$ since $f'' (x)$ is negative

Concave up on $(1, \infty)$ since $f'' (x)$ is positive

Step 8

Find the Concavity f(x)=9x^4-54x^2

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