Calculus Examples

$f (x) = 10 x^{6} - 13 x^{5}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $10 x^{6} - 13 x^{5}$ with respect to $x$ is $\frac{d}{d x} [10 x^{6}] + \frac{d}{d x} [- 13 x^{5}]$ .

$f' (x) = \frac{d}{d x} (10 x^{6}) + \frac{d}{d x} (- 13 x^{5})$

Step 1.1.2

Evaluate $\frac{d}{d x} [10 x^{6}]$ .

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Step 1.1.2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{6}$ with respect to $x$ is $10 \frac{d}{d x} [x^{6}]$ .

$f' (x) = 10 \frac{d}{d x} (x^{6}) + \frac{d}{d x} (- 13 x^{5})$

Step 1.1.2.2

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

$f' (x) = 10 (6 x^{5}) + \frac{d}{d x} (- 13 x^{5})$

Step 1.1.2.3

Multiply $6$ by $10$ .

$f' (x) = 60 x^{5} + \frac{d}{d x} (- 13 x^{5})$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 13 x^{5}]$ .

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Step 1.1.3.1

Since $- 13$ is constant with respect to $x$ , the derivative of $- 13 x^{5}$ with respect to $x$ is $- 13 \frac{d}{d x} [x^{5}]$ .

$f' (x) = 60 x^{5} - 13 \frac{d}{d x} x^{5}$

Step 1.1.3.2

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

$f' (x) = 60 x^{5} - 13 (5 x^{4})$

Step 1.1.3.3

Multiply $5$ by $- 13$ .

$f' (x) = 60 x^{5} - 65 x^{4}$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $60 x^{5} - 65 x^{4}$ with respect to $x$ is $\frac{d}{d x} [60 x^{5}] + \frac{d}{d x} [- 65 x^{4}]$ .

$f'' (x) = \frac{d}{d x} (60 x^{5}) + \frac{d}{d x} (- 65 x^{4})$

Step 1.2.2

Evaluate $\frac{d}{d x} [60 x^{5}]$ .

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Step 1.2.2.1

Since $60$ is constant with respect to $x$ , the derivative of $60 x^{5}$ with respect to $x$ is $60 \frac{d}{d x} [x^{5}]$ .

$f'' (x) = 60 \frac{d}{d x} (x^{5}) + \frac{d}{d x} (- 65 x^{4})$

Step 1.2.2.2

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

$f'' (x) = 60 (5 x^{4}) + \frac{d}{d x} (- 65 x^{4})$

Step 1.2.2.3

Multiply $5$ by $60$ .

$f'' (x) = 300 x^{4} + \frac{d}{d x} (- 65 x^{4})$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 65 x^{4}]$ .

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Step 1.2.3.1

Since $- 65$ is constant with respect to $x$ , the derivative of $- 65 x^{4}$ with respect to $x$ is $- 65 \frac{d}{d x} [x^{4}]$ .

$f'' (x) = 300 x^{4} - 65 \frac{d}{d x} x^{4}$

Step 1.2.3.2

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

$f'' (x) = 300 x^{4} - 65 (4 x^{3})$

Step 1.2.3.3

Multiply $4$ by $- 65$ .

$f'' (x) = 300 x^{4} - 260 x^{3}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $300 x^{4} - 260 x^{3}$ .

$300 x^{4} - 260 x^{3}$

Step 2

Set the second derivative equal to $0$ then solve the equation $300 x^{4} - 260 x^{3} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$300 x^{4} - 260 x^{3} = 0$

Step 2.2

Factor $20 x^{3}$ out of $300 x^{4} - 260 x^{3}$ .

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Step 2.2.1

Factor $20 x^{3}$ out of $300 x^{4}$ .

$20 x^{3} (15 x) - 260 x^{3} = 0$

Step 2.2.2

Factor $20 x^{3}$ out of $- 260 x^{3}$ .

$20 x^{3} (15 x) + 20 x^{3} (- 13) = 0$

Step 2.2.3

Factor $20 x^{3}$ out of $20 x^{3} (15 x) + 20 x^{3} (- 13)$ .

$20 x^{3} (15 x - 13) = 0$

Step 2.3

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

$x^{3} = 0$

$15 x - 13 = 0$

Step 2.4

Set $x^{3}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Step 2.4.2

Solve $x^{3} = 0$ for $x$ .

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Step 2.4.2.1

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

$x = \sqrt[3]{0}$

Step 2.4.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.4.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 2.4.2.2.2

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

$x = 0$

Step 2.5

Set $15 x - 13$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $15 x - 13$ equal to $0$ .

$15 x - 13 = 0$

Step 2.5.2

Solve $15 x - 13 = 0$ for $x$ .

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Step 2.5.2.1

Add $13$ to both sides of the equation.

$15 x = 13$

Step 2.5.2.2

Divide each term in $15 x = 13$ by $15$ and simplify.

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Step 2.5.2.2.1

Divide each term in $15 x = 13$ by $15$ .

$\frac{15 x}{15} = \frac{13}{15}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $15$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{15 x}{15} = \frac{13}{15}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{13}{15}$

Step 2.6

The final solution is all the values that make $20 x^{3} (15 x - 13) = 0$ true.

$x = 0, \frac{13}{15}$

Step 3

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

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Step 3.1

Substitute $0$ in $f (x) = 10 x^{6} - 13 x^{5}$ to find the value of $y$ .

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Step 3.1.1

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

$f (0) = 10 {(0)}^{6} - 13 {(0)}^{5}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

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

$f (0) = 10 \cdot 0 - 13 {(0)}^{5}$

Step 3.1.2.1.2

Multiply $10$ by $0$ .

$f (0) = 0 - 13 {(0)}^{5}$

Step 3.1.2.1.3

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

$f (0) = 0 - 13 \cdot 0$

Step 3.1.2.1.4

Multiply $- 13$ by $0$ .

$f (0) = 0 + 0$

Step 3.1.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 3.1.2.3

The final answer is $0$ .

$0$

Step 3.2

The point found by substituting $0$ in $f (x) = 10 x^{6} - 13 x^{5}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 3.3

Substitute $\frac{13}{15}$ in $f (x) = 10 x^{6} - 13 x^{5}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $\frac{13}{15}$ in the expression.

$f (\frac{13}{15}) = 10 {(\frac{13}{15})}^{6} - 13 {(\frac{13}{15})}^{5}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Apply the product rule to $\frac{13}{15}$ .

$f (\frac{13}{15}) = 10 (\frac{13^{6}}{15^{6}}) - 13 {(\frac{13}{15})}^{5}$

Step 3.3.2.1.2

Raise $13$ to the power of $6$ .

$f (\frac{13}{15}) = 10 (\frac{4826809}{15^{6}}) - 13 {(\frac{13}{15})}^{5}$

Step 3.3.2.1.3

Raise $15$ to the power of $6$ .

$f (\frac{13}{15}) = 10 (\frac{4826809}{11390625}) - 13 {(\frac{13}{15})}^{5}$

Step 3.3.2.1.4

Cancel the common factor of $5$ .

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Step 3.3.2.1.4.1

Factor $5$ out of $10$ .

$f (\frac{13}{15}) = 5 (2) (\frac{4826809}{11390625}) - 13 {(\frac{13}{15})}^{5}$

Step 3.3.2.1.4.2

Factor $5$ out of $11390625$ .

$f (\frac{13}{15}) = 5 \cdot (2 (\frac{4826809}{5 \cdot 2278125})) - 13 {(\frac{13}{15})}^{5}$

Step 3.3.2.1.4.3

Cancel the common factor.

$f (\frac{13}{15}) = 5 \cdot (2 (\frac{4826809}{5 \cdot 2278125})) - 13 {(\frac{13}{15})}^{5}$

Step 3.3.2.1.4.4

Rewrite the expression.

$f (\frac{13}{15}) = 2 (\frac{4826809}{2278125}) - 13 {(\frac{13}{15})}^{5}$

Step 3.3.2.1.5

Combine $2$ and $\frac{4826809}{2278125}$ .

$f (\frac{13}{15}) = \frac{2 \cdot 4826809}{2278125} - 13 {(\frac{13}{15})}^{5}$

Step 3.3.2.1.6

Multiply $2$ by $4826809$ .

$f (\frac{13}{15}) = \frac{9653618}{2278125} - 13 {(\frac{13}{15})}^{5}$

Step 3.3.2.1.7

Apply the product rule to $\frac{13}{15}$ .

$f (\frac{13}{15}) = \frac{9653618}{2278125} - 13 \frac{13^{5}}{15^{5}}$

Step 3.3.2.1.8

Raise $13$ to the power of $5$ .

$f (\frac{13}{15}) = \frac{9653618}{2278125} - 13 \frac{371293}{15^{5}}$

Step 3.3.2.1.9

Raise $15$ to the power of $5$ .

$f (\frac{13}{15}) = \frac{9653618}{2278125} - 13 (\frac{371293}{759375})$

Step 3.3.2.1.10

Multiply $- 13 (\frac{371293}{759375})$ .

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Step 3.3.2.1.10.1

Combine $- 13$ and $\frac{371293}{759375}$ .

$f (\frac{13}{15}) = \frac{9653618}{2278125} + \frac{- 13 \cdot 371293}{759375}$

Step 3.3.2.1.10.2

Multiply $- 13$ by $371293$ .

$f (\frac{13}{15}) = \frac{9653618}{2278125} + \frac{- 4826809}{759375}$

Step 3.3.2.1.11

Move the negative in front of the fraction.

$f (\frac{13}{15}) = \frac{9653618}{2278125} - \frac{4826809}{759375}$

Step 3.3.2.2

To write $- \frac{4826809}{759375}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{13}{15}) = \frac{9653618}{2278125} - \frac{4826809}{759375} \cdot \frac{3}{3}$

Step 3.3.2.3

Write each expression with a common denominator of $2278125$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.3.1

Multiply $\frac{4826809}{759375}$ by $\frac{3}{3}$ .

$f (\frac{13}{15}) = \frac{9653618}{2278125} - \frac{4826809 \cdot 3}{759375 \cdot 3}$

Step 3.3.2.3.2

Multiply $759375$ by $3$ .

$f (\frac{13}{15}) = \frac{9653618}{2278125} - \frac{4826809 \cdot 3}{2278125}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$f (\frac{13}{15}) = \frac{9653618 - 4826809 \cdot 3}{2278125}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $- 4826809$ by $3$ .

$f (\frac{13}{15}) = \frac{9653618 - 14480427}{2278125}$

Step 3.3.2.5.2

Subtract $14480427$ from $9653618$ .

$f (\frac{13}{15}) = \frac{- 4826809}{2278125}$

Step 3.3.2.6

Move the negative in front of the fraction.

$f (\frac{13}{15}) = - \frac{4826809}{2278125}$

Step 3.3.2.7

The final answer is $- \frac{4826809}{2278125}$ .

$- \frac{4826809}{2278125}$

Step 3.4

The point found by substituting $\frac{13}{15}$ in $f (x) = 10 x^{6} - 13 x^{5}$ is $(\frac{13}{15}, - \frac{4826809}{2278125})$ . This point can be an inflection point.

$(\frac{13}{15}, - \frac{4826809}{2278125})$

Step 3.5

Determine the points that could be inflection points.

$(0, 0), (\frac{13}{15}, - \frac{4826809}{2278125})$

Step 4

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

$(- \infty, 0) \cup (0, \frac{13}{15}) \cup (\frac{13}{15}, \infty)$

Step 5

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

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Step 5.1

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

$f'' (- 0.1) = 300 {(- 0.1)}^{4} - 260 {(- 0.1)}^{3}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f'' (- 0.1) = 300 \cdot 0.0001 - 260 {(- 0.1)}^{3}$

Step 5.2.1.2

Multiply $300$ by $0.0001$ .

$f'' (- 0.1) = 0.03 - 260 {(- 0.1)}^{3}$

Step 5.2.1.3

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

$f'' (- 0.1) = 0.03 - 260 \cdot - 0.001$

Step 5.2.1.4

Multiply $- 260$ by $- 0.001$ .

$f'' (- 0.1) = 0.03 + 0.26$

Step 5.2.2

Add $0.03$ and $0.26$ .

$f'' (- 0.1) = 0.29$

Step 5.2.3

The final answer is $0.29$ .

$0.29$

Step 5.3

At $- 0.1$ , the second derivative is $0.29$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 0)$ .

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

Step 6

Substitute a value from the interval $(0, \frac{13}{15})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0.4\overline{3}$ in the expression.

$f'' (0.4\overline{3}) = 300 {(0.4\overline{3})}^{4} - 260 {(0.4\overline{3})}^{3}$

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 $0.4\overline{3}$ to the power of $4$ .

$f'' (0.4\overline{3}) = 300 \cdot 0.03526049 - 260 {(0.4\overline{3})}^{3}$

Step 6.2.1.2

Multiply $300$ by $0.03526049$ .

$f'' (0.4\overline{3}) = 10.57\overline{814} - 260 {(0.4\overline{3})}^{3}$

Step 6.2.1.3

Raise $0.4\overline{3}$ to the power of $3$ .

$f'' (0.4\overline{3}) = 10.57\overline{814} - 260 \cdot 0.081\overline{370}$

Step 6.2.1.4

Multiply $- 260$ by $0.081\overline{370}$ .

$f'' (0.4\overline{3}) = 10.57\overline{814} - 21.15629629$

Step 6.2.2

Subtract $21.15629629$ from $10.57\overline{814}$ .

$f'' (0.4\overline{3}) = - 10.57814814$

Step 6.2.3

The final answer is $- 10.57814814$ .

$- 10.57814814$

Step 6.3

At $0.4\overline{3}$ , the second derivative is $- 10.57814814$ . Since this is negative, the second derivative is decreasing on the interval $(0, \frac{13}{15})$

Decreasing on $(0, \frac{13}{15})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(\frac{13}{15}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.9\overline{6}$ in the expression.

$f'' (0.9\overline{6}) = 300 {(0.9\overline{6})}^{4} - 260 {(0.9\overline{6})}^{3}$

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 $0.9\overline{6}$ to the power of $4$ .

$f'' (0.9\overline{6}) = 300 \cdot 0.87318641 - 260 {(0.9\overline{6})}^{3}$

Step 7.2.1.2

Multiply $300$ by $0.87318641$ .

$f'' (0.9\overline{6}) = 261.95592592 - 260 {(0.9\overline{6})}^{3}$

Step 7.2.1.3

Raise $0.9\overline{6}$ to the power of $3$ .

$f'' (0.9\overline{6}) = 261.95592592 - 260 \cdot 0.903\overline{296}$

Step 7.2.1.4

Multiply $- 260$ by $0.903\overline{296}$ .

$f'' (0.9\overline{6}) = 261.95592592 - 234.85703703$

Step 7.2.2

Subtract $234.85703703$ from $261.95592592$ .

$f'' (0.9\overline{6}) = 27.09888888$

Step 7.2.3

The final answer is $27.09888888$ .

$27.09888888$

Step 7.3

At $0.9\overline{6}$ , the second derivative is $27.09888888$ . Since this is positive, the second derivative is increasing on the interval $(\frac{13}{15}, \infty)$ .

Increasing on $(\frac{13}{15}, \infty)$ since $f'' (x) > 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(0, 0), (\frac{13}{15}, - \frac{4826809}{2278125})$ .

$(0, 0), (\frac{13}{15}, - \frac{4826809}{2278125})$

Step 9