Calculus Examples

$\frac{1}{6} x^{6} + x^{5} + \frac{5}{3} x^{4}$

Step 1

Write $\frac{1}{6} x^{6} + x^{5} + \frac{5}{3} x^{4}$ as a function.

$f (x) = \frac{1}{6} x^{6} + x^{5} + \frac{5}{3} x^{4}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $\frac{1}{6} x^{6} + x^{5} + \frac{5}{3} x^{4}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{6} x^{6}] + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{5}{3} x^{4}]$ .

$\frac{d}{d x} [\frac{1}{6} x^{6}] + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{5}{3} x^{4}]$

Step 2.1.2

Evaluate $\frac{d}{d x} [\frac{1}{6} x^{6}]$ .

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Step 2.1.2.1

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{1}{6} x^{6}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [x^{6}]$ .

$\frac{1}{6} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{5}{3} x^{4}]$

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$\frac{1}{6} (6 x^{5}) + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{5}{3} x^{4}]$

Step 2.1.2.3

Combine $6$ and $\frac{1}{6}$ .

$\frac{6}{6} x^{5} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{5}{3} x^{4}]$

Step 2.1.2.4

Combine $\frac{6}{6}$ and $x^{5}$ .

$\frac{6 x^{5}}{6} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{5}{3} x^{4}]$

Step 2.1.2.5

Cancel the common factor of $6$ .

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Step 2.1.2.5.1

Cancel the common factor.

$\frac{6 x^{5}}{6} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{5}{3} x^{4}]$

Step 2.1.2.5.2

Divide $x^{5}$ by $1$ .

$x^{5} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{5}{3} x^{4}]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$x^{5} + 5 x^{4} + \frac{d}{d x} [\frac{5}{3} x^{4}]$

Step 2.1.4

Evaluate $\frac{d}{d x} [\frac{5}{3} x^{4}]$ .

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Step 2.1.4.1

Since $\frac{5}{3}$ is constant with respect to $x$ , the derivative of $\frac{5}{3} x^{4}$ with respect to $x$ is $\frac{5}{3} \frac{d}{d x} [x^{4}]$ .

$x^{5} + 5 x^{4} + \frac{5}{3} \frac{d}{d x} [x^{4}]$

Step 2.1.4.2

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

$x^{5} + 5 x^{4} + \frac{5}{3} (4 x^{3})$

Step 2.1.4.3

Combine $4$ and $\frac{5}{3}$ .

$x^{5} + 5 x^{4} + \frac{4 \cdot 5}{3} x^{3}$

Step 2.1.4.4

Multiply $4$ by $5$ .

$x^{5} + 5 x^{4} + \frac{20}{3} x^{3}$

Step 2.1.4.5

Combine $\frac{20}{3}$ and $x^{3}$ .

$f' (x) = x^{5} + 5 x^{4} + \frac{20 x^{3}}{3}$

Step 2.2

Find the second derivative.

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Step 2.2.1

Differentiate.

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Step 2.2.1.1

By the Sum Rule, the derivative of $x^{5} + 5 x^{4} + \frac{20 x^{3}}{3}$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [\frac{20 x^{3}}{3}]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [\frac{20 x^{3}}{3}]$

Step 2.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 x^{4} + \frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [\frac{20 x^{3}}{3}]$

Step 2.2.2

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

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Step 2.2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$5 x^{4} + 5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{20 x^{3}}{3}]$

Step 2.2.2.2

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

$5 x^{4} + 5 (4 x^{3}) + \frac{d}{d x} [\frac{20 x^{3}}{3}]$

Step 2.2.2.3

Multiply $4$ by $5$ .

$5 x^{4} + 20 x^{3} + \frac{d}{d x} [\frac{20 x^{3}}{3}]$

Step 2.2.3

Evaluate $\frac{d}{d x} [\frac{20 x^{3}}{3}]$ .

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Step 2.2.3.1

Since $\frac{20}{3}$ is constant with respect to $x$ , the derivative of $\frac{20 x^{3}}{3}$ with respect to $x$ is $\frac{20}{3} \frac{d}{d x} [x^{3}]$ .

$5 x^{4} + 20 x^{3} + \frac{20}{3} \frac{d}{d x} [x^{3}]$

Step 2.2.3.2

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

$5 x^{4} + 20 x^{3} + \frac{20}{3} (3 x^{2})$

Step 2.2.3.3

Combine $3$ and $\frac{20}{3}$ .

$5 x^{4} + 20 x^{3} + \frac{3 \cdot 20}{3} x^{2}$

Step 2.2.3.4

Multiply $3$ by $20$ .

$5 x^{4} + 20 x^{3} + \frac{60}{3} x^{2}$

Step 2.2.3.5

Combine $\frac{60}{3}$ and $x^{2}$ .

$5 x^{4} + 20 x^{3} + \frac{60 x^{2}}{3}$

Step 2.2.3.6

Cancel the common factor of $60$ and $3$ .

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Step 2.2.3.6.1

Factor $3$ out of $60 x^{2}$ .

$5 x^{4} + 20 x^{3} + \frac{3 (20 x^{2})}{3}$

Step 2.2.3.6.2

Cancel the common factors.

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Step 2.2.3.6.2.1

Factor $3$ out of $3$ .

$5 x^{4} + 20 x^{3} + \frac{3 (20 x^{2})}{3 (1)}$

Step 2.2.3.6.2.2

Cancel the common factor.

$5 x^{4} + 20 x^{3} + \frac{3 (20 x^{2})}{3 \cdot 1}$

Step 2.2.3.6.2.3

Rewrite the expression.

$5 x^{4} + 20 x^{3} + \frac{20 x^{2}}{1}$

Step 2.2.3.6.2.4

Divide $20 x^{2}$ by $1$ .

$f'' (x) = 5 x^{4} + 20 x^{3} + 20 x^{2}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $5 x^{4} + 20 x^{3} + 20 x^{2}$ .

$5 x^{4} + 20 x^{3} + 20 x^{2}$

Step 3

Set the second derivative equal to $0$ then solve the equation $5 x^{4} + 20 x^{3} + 20 x^{2} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$5 x^{4} + 20 x^{3} + 20 x^{2} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $5 x^{2}$ out of $5 x^{4} + 20 x^{3} + 20 x^{2}$ .

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Step 3.2.1.1

Factor $5 x^{2}$ out of $5 x^{4}$ .

$5 x^{2} (x^{2}) + 20 x^{3} + 20 x^{2} = 0$

Step 3.2.1.2

Factor $5 x^{2}$ out of $20 x^{3}$ .

$5 x^{2} (x^{2}) + 5 x^{2} (4 x) + 20 x^{2} = 0$

Step 3.2.1.3

Factor $5 x^{2}$ out of $20 x^{2}$ .

$5 x^{2} (x^{2}) + 5 x^{2} (4 x) + 5 x^{2} (4) = 0$

Step 3.2.1.4

Factor $5 x^{2}$ out of $5 x^{2} (x^{2}) + 5 x^{2} (4 x)$ .

$5 x^{2} (x^{2} + 4 x) + 5 x^{2} (4) = 0$

Step 3.2.1.5

Factor $5 x^{2}$ out of $5 x^{2} (x^{2} + 4 x) + 5 x^{2} (4)$ .

$5 x^{2} (x^{2} + 4 x + 4) = 0$

Step 3.2.2

Factor using the perfect square rule.

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Step 3.2.2.1

Rewrite $4$ as $2^{2}$ .

$5 x^{2} (x^{2} + 4 x + 2^{2}) = 0$

Step 3.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 3.2.2.3

Rewrite the polynomial.

$5 x^{2} (x^{2} + 2 \cdot x \cdot 2 + 2^{2}) = 0$

Step 3.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 2$ .

$5 x^{2} {(x + 2)}^{2} = 0$

Step 3.3

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

$x^{2} = 0$

${(x + 2)}^{2} = 0$

Step 3.4

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

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Step 3.4.1

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

$x^{2} = 0$

Step 3.4.2

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

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Step 3.4.2.1

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

$x = \pm \sqrt{0}$

Step 3.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.4.2.2.1

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

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

Step 3.4.2.2.2

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

$x = \pm 0$

Step 3.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3.5

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

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Step 3.5.1

Set ${(x + 2)}^{2}$ equal to $0$ .

${(x + 2)}^{2} = 0$

Step 3.5.2

Solve ${(x + 2)}^{2} = 0$ for $x$ .

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Step 3.5.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.5.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.6

The final solution is all the values that make $5 x^{2} {(x + 2)}^{2} = 0$ true.

$x = 0, - 2$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $0$ in $f (x) = \frac{1}{6} x^{6} + x^{5} + \frac{5}{3} x^{4}$ to find the value of $y$ .

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Step 4.1.1

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

$f (0) = \frac{1}{6} \cdot {(0)}^{6} + {(0)}^{5} + \frac{5}{3} \cdot {(0)}^{4}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$f (0) = \frac{1}{6} \cdot 0 + {(0)}^{5} + \frac{5}{3} \cdot {(0)}^{4}$

Step 4.1.2.1.2

Multiply $\frac{1}{6}$ by $0$ .

$f (0) = 0 + {(0)}^{5} + \frac{5}{3} \cdot {(0)}^{4}$

Step 4.1.2.1.3

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

$f (0) = 0 + 0 + \frac{5}{3} \cdot {(0)}^{4}$

Step 4.1.2.1.4

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

$f (0) = 0 + 0 + \frac{5}{3} \cdot 0$

Step 4.1.2.1.5

Multiply $\frac{5}{3}$ by $0$ .

$f (0) = 0 + 0 + 0$

Step 4.1.2.2

Simplify by adding numbers.

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Step 4.1.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 4.1.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 4.1.2.3

The final answer is $0$ .

$0$

Step 4.2

The point found by substituting $0$ in $f (x) = \frac{1}{6} x^{6} + x^{5} + \frac{5}{3} x^{4}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 4.3

Substitute $- 2$ in $f (x) = \frac{1}{6} x^{6} + x^{5} + \frac{5}{3} x^{4}$ to find the value of $y$ .

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Step 4.3.1

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

$f (- 2) = \frac{1}{6} \cdot {(- 2)}^{6} + {(- 2)}^{5} + \frac{5}{3} \cdot {(- 2)}^{4}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

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

$f (- 2) = \frac{1}{6} \cdot 64 + {(- 2)}^{5} + \frac{5}{3} \cdot {(- 2)}^{4}$

Step 4.3.2.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.1.2.1

Factor $2$ out of $6$ .

$f (- 2) = \frac{1}{2 (3)} \cdot 64 + {(- 2)}^{5} + \frac{5}{3} \cdot {(- 2)}^{4}$

Step 4.3.2.1.2.2

Factor $2$ out of $64$ .

$f (- 2) = \frac{1}{2 \cdot 3} \cdot (2 \cdot 32) + {(- 2)}^{5} + \frac{5}{3} \cdot {(- 2)}^{4}$

Step 4.3.2.1.2.3

Cancel the common factor.

$f (- 2) = \frac{1}{2 \cdot 3} \cdot (2 \cdot 32) + {(- 2)}^{5} + \frac{5}{3} \cdot {(- 2)}^{4}$

Step 4.3.2.1.2.4

Rewrite the expression.

$f (- 2) = \frac{1}{3} \cdot 32 + {(- 2)}^{5} + \frac{5}{3} \cdot {(- 2)}^{4}$

Step 4.3.2.1.3

Combine $\frac{1}{3}$ and $32$ .

$f (- 2) = \frac{32}{3} + {(- 2)}^{5} + \frac{5}{3} \cdot {(- 2)}^{4}$

Step 4.3.2.1.4

Raise $- 2$ to the power of $5$ .

$f (- 2) = \frac{32}{3} - 32 + \frac{5}{3} \cdot {(- 2)}^{4}$

Step 4.3.2.1.5

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

$f (- 2) = \frac{32}{3} - 32 + \frac{5}{3} \cdot 16$

Step 4.3.2.1.6

Multiply $\frac{5}{3} \cdot 16$ .

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Step 4.3.2.1.6.1

Combine $\frac{5}{3}$ and $16$ .

$f (- 2) = \frac{32}{3} - 32 + \frac{5 \cdot 16}{3}$

Step 4.3.2.1.6.2

Multiply $5$ by $16$ .

$f (- 2) = \frac{32}{3} - 32 + \frac{80}{3}$

Step 4.3.2.2

Combine fractions.

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Step 4.3.2.2.1

Combine the numerators over the common denominator.

$f (- 2) = - 32 + \frac{32 + 80}{3}$

Step 4.3.2.2.2

Add $32$ and $80$ .

$f (- 2) = - 32 + \frac{112}{3}$

Step 4.3.2.3

To write $- 32$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- 2) = - 32 \cdot \frac{3}{3} + \frac{112}{3}$

Step 4.3.2.4

Combine $- 32$ and $\frac{3}{3}$ .

$f (- 2) = \frac{- 32 \cdot 3}{3} + \frac{112}{3}$

Step 4.3.2.5

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 32 \cdot 3 + 112}{3}$

Step 4.3.2.6

Simplify the numerator.

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Step 4.3.2.6.1

Multiply $- 32$ by $3$ .

$f (- 2) = \frac{- 96 + 112}{3}$

Step 4.3.2.6.2

Add $- 96$ and $112$ .

$f (- 2) = \frac{16}{3}$

Step 4.3.2.7

The final answer is $\frac{16}{3}$ .

$\frac{16}{3}$

Step 4.4

The point found by substituting $- 2$ in $f (x) = \frac{1}{6} x^{6} + x^{5} + \frac{5}{3} x^{4}$ is $(- 2, \frac{16}{3})$ . This point can be an inflection point.

$(- 2, \frac{16}{3})$

Step 4.5

Determine the points that could be inflection points.

$(0, 0), (- 2, \frac{16}{3})$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - 2) \cup (- 2, 0) \cup (0, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 2)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 2.1$ in the expression.

$f'' (- 2.1) = 5 {(- 2.1)}^{4} + 20 {(- 2.1)}^{3} + 20 {(- 2.1)}^{2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 2.1$ to the power of $4$ .

$f'' (- 2.1) = 5 \cdot 19.4481 + 20 {(- 2.1)}^{3} + 20 {(- 2.1)}^{2}$

Step 6.2.1.2

Multiply $5$ by $19.4481$ .

$f'' (- 2.1) = 97.2405 + 20 {(- 2.1)}^{3} + 20 {(- 2.1)}^{2}$

Step 6.2.1.3

Raise $- 2.1$ to the power of $3$ .

$f'' (- 2.1) = 97.2405 + 20 \cdot - 9.261 + 20 {(- 2.1)}^{2}$

Step 6.2.1.4

Multiply $20$ by $- 9.261$ .

$f'' (- 2.1) = 97.2405 - 185.22 + 20 {(- 2.1)}^{2}$

Step 6.2.1.5

Raise $- 2.1$ to the power of $2$ .

$f'' (- 2.1) = 97.2405 - 185.22 + 20 \cdot 4.41$

Step 6.2.1.6

Multiply $20$ by $4.41$ .

$f'' (- 2.1) = 97.2405 - 185.22 + 88.2$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $185.22$ from $97.2405$ .

$f'' (- 2.1) = - 87.9795 + 88.2$

Step 6.2.2.2

Add $- 87.9795$ and $88.2$ .

$f'' (- 2.1) = 0.2205$

Step 6.2.3

The final answer is $0.2205$ .

$0.2205$

Step 6.3

At $- 2.1$ , the second derivative is $0.2205$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 2)$ .

Increasing on $(- \infty, - 2)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(- 2, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 1$ in the expression.

$f'' (- 1) = 5 {(- 1)}^{4} + 20 {(- 1)}^{3} + 20 {(- 1)}^{2}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $- 1$ to the power of $4$ .

$f'' (- 1) = 5 \cdot 1 + 20 {(- 1)}^{3} + 20 {(- 1)}^{2}$

Step 7.2.1.2

Multiply $5$ by $1$ .

$f'' (- 1) = 5 + 20 {(- 1)}^{3} + 20 {(- 1)}^{2}$

Step 7.2.1.3

Raise $- 1$ to the power of $3$ .

$f'' (- 1) = 5 + 20 \cdot - 1 + 20 {(- 1)}^{2}$

Step 7.2.1.4

Multiply $20$ by $- 1$ .

$f'' (- 1) = 5 - 20 + 20 {(- 1)}^{2}$

Step 7.2.1.5

Raise $- 1$ to the power of $2$ .

$f'' (- 1) = 5 - 20 + 20 \cdot 1$

Step 7.2.1.6

Multiply $20$ by $1$ .

$f'' (- 1) = 5 - 20 + 20$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $20$ from $5$ .

$f'' (- 1) = - 15 + 20$

Step 7.2.2.2

Add $- 15$ and $20$ .

$f'' (- 1) = 5$

Step 7.2.3

The final answer is $5$ .

$5$

Step 7.3

At $- 1$ , the second derivative is $5$ . Since this is positive, the second derivative is increasing on the interval $(- 2, 0)$ .

Increasing on $(- 2, 0)$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(0, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $0.1$ in the expression.

$f'' (0.1) = 5 {(0.1)}^{4} + 20 {(0.1)}^{3} + 20 {(0.1)}^{2}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $0.1$ to the power of $4$ .

$f'' (0.1) = 5 \cdot 0.0001 + 20 {(0.1)}^{3} + 20 {(0.1)}^{2}$

Step 8.2.1.2

Multiply $5$ by $0.0001$ .

$f'' (0.1) = 0.0005 + 20 {(0.1)}^{3} + 20 {(0.1)}^{2}$

Step 8.2.1.3

Raise $0.1$ to the power of $3$ .

$f'' (0.1) = 0.0005 + 20 \cdot 0.001 + 20 {(0.1)}^{2}$

Step 8.2.1.4

Multiply $20$ by $0.001$ .

$f'' (0.1) = 0.0005 + 0.02 + 20 {(0.1)}^{2}$

Step 8.2.1.5

Raise $0.1$ to the power of $2$ .

$f'' (0.1) = 0.0005 + 0.02 + 20 \cdot 0.01$

Step 8.2.1.6

Multiply $20$ by $0.01$ .

$f'' (0.1) = 0.0005 + 0.02 + 0.2$

Step 8.2.2

Simplify by adding numbers.

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Step 8.2.2.1

Add $0.0005$ and $0.02$ .

$f'' (0.1) = 0.0205 + 0.2$

Step 8.2.2.2

Add $0.0205$ and $0.2$ .

$f'' (0.1) = 0.2205$

Step 8.2.3

The final answer is $0.2205$ .

$0.2205$

Step 8.3

At $0.1$ , the second derivative is $0.2205$ . Since this is positive, the second derivative is increasing on the interval $(0, \infty)$ .

Increasing on $(0, \infty)$ since $f'' (x) > 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. There are no points on the graph that satisfy these requirements.

No Inflection Points