Calculus Examples

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

Step 1

Write $x^{4} - 2 x^{2} + 3$ as a function.

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

Step 2

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

Tap for more steps...

Step 2.1

Find the second derivative.

Tap for more steps...

Step 2.1.1

Find the first derivative.

Tap for more steps...

Step 2.1.1.1

Differentiate.

Tap for more steps...

Step 2.1.1.1.1

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

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

Step 2.1.1.1.2

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

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

Step 2.1.1.2

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

Tap for more steps...

Step 2.1.1.2.1

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

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

Step 2.1.1.2.2

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

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

Step 2.1.1.2.3

Multiply $2$ by $- 2$ .

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

Step 2.1.1.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 2.1.1.3.2

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

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

Step 2.1.2

Find the second derivative.

Tap for more steps...

Step 2.1.2.1

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

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

Step 2.1.2.2

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

Tap for more steps...

Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

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

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

Step 2.1.2.2.3

Multiply $3$ by $4$ .

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

Step 2.1.2.3

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

Tap for more steps...

Step 2.1.2.3.1

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

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

Step 2.1.2.3.2

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

$12 x^{2} - 4 \cdot 1$

Step 2.1.2.3.3

Multiply $- 4$ by $1$ .

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

Step 2.1.3

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

$12 x^{2} - 4$

Step 2.2

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

Tap for more steps...

Step 2.2.1

Set the second derivative equal to $0$ .

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

Step 2.2.2

Add $4$ to both sides of the equation.

$12 x^{2} = 4$

Step 2.2.3

Divide each term in $12 x^{2} = 4$ by $12$ and simplify.

Tap for more steps...

Step 2.2.3.1

Divide each term in $12 x^{2} = 4$ by $12$ .

$\frac{12 x^{2}}{12} = \frac{4}{12}$

Step 2.2.3.2

Simplify the left side.

Tap for more steps...

Step 2.2.3.2.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{12 x^{2}}{12} = \frac{4}{12}$

Step 2.2.3.2.1.2

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

$x^{2} = \frac{4}{12}$

Step 2.2.3.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.3.1

Cancel the common factor of $4$ and $12$ .

Tap for more steps...

Step 2.2.3.3.1.1

Factor $4$ out of $4$ .

$x^{2} = \frac{4 (1)}{12}$

Step 2.2.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.2.3.3.1.2.1

Factor $4$ out of $12$ .

$x^{2} = \frac{4 \cdot 1}{4 \cdot 3}$

Step 2.2.3.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{4 \cdot 1}{4 \cdot 3}$

Step 2.2.3.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{1}{3}$

Step 2.2.4

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

$x = \pm \sqrt{\frac{1}{3}}$

Step 2.2.5

Simplify $\pm \sqrt{\frac{1}{3}}$ .

Tap for more steps...

Step 2.2.5.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{3}}$

Step 2.2.5.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{3}}$

Step 2.2.5.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.2.5.4

Combine and simplify the denominator.

Tap for more steps...

Step 2.2.5.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.2.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.2.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.2.5.4.4

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

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.2.5.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.2.5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 2.2.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.2.5.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.2.5.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.2.5.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.2.5.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{3}}{3^{1}}$

Step 2.2.5.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{3}}{3}$

Step 2.2.6

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

Tap for more steps...

Step 2.2.6.1

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

$x = \frac{\sqrt{3}}{3}$

Step 2.2.6.2

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

$x = - \frac{\sqrt{3}}{3}$

Step 2.2.6.3

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

$x = \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Step 3

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 4

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

$(- \infty, - \frac{\sqrt{3}}{3}) \cup (- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}) \cup (\frac{\sqrt{3}}{3}, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - \frac{\sqrt{3}}{3})$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $- 3$ in the expression.

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

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

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

$f'' (- 3) = 12 \cdot 9 - 4$

Step 5.2.1.2

Multiply $12$ by $9$ .

$f'' (- 3) = 108 - 4$

Step 5.2.2

Subtract $4$ from $108$ .

$f'' (- 3) = 104$

Step 5.2.3

The final answer is $104$ .

$104$

Step 5.3

The graph is concave up on the interval $(- \infty, - \frac{\sqrt{3}}{3})$ because $f'' (- 3)$ is positive.

Concave up on $(- \infty, - \frac{\sqrt{3}}{3})$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 6.1

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

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

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

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

$f'' (0) = 12 \cdot 0 - 4$

Step 6.2.1.2

Multiply $12$ by $0$ .

$f'' (0) = 0 - 4$

Step 6.2.2

Subtract $4$ from $0$ .

$f'' (0) = - 4$

Step 6.2.3

The final answer is $- 4$ .

$- 4$

Step 6.3

The graph is concave down on the interval $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ because $f'' (0)$ is negative.

Concave down on $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(\frac{\sqrt{3}}{3}, \infty)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $3$ in the expression.

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

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1

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

$f'' (3) = 12 \cdot 9 - 4$

Step 7.2.1.2

Multiply $12$ by $9$ .

$f'' (3) = 108 - 4$

Step 7.2.2

Subtract $4$ from $108$ .

$f'' (3) = 104$

Step 7.2.3

The final answer is $104$ .

$104$

Step 7.3

The graph is concave up on the interval $(\frac{\sqrt{3}}{3}, \infty)$ because $f'' (3)$ is positive.

Concave up on $(\frac{\sqrt{3}}{3}, \infty)$ since $f'' (x)$ is positive

Step 8

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

Concave up on $(- \infty, - \frac{\sqrt{3}}{3})$ since $f'' (x)$ is positive

Concave down on $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ since $f'' (x)$ is negative

Concave up on $(\frac{\sqrt{3}}{3}, \infty)$ since $f'' (x)$ is positive

Step 9