Calculus Examples

$4 x^{3} - 6 x^{2} - 24 x$

Step 1

Write $4 x^{3} - 6 x^{2} - 24 x$ as a function.

$f (x) = 4 x^{3} - 6 x^{2} - 24 x$

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 $4 x^{3} - 6 x^{2} - 24 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 24 x]$ .

$f' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (- 24 x)$

Step 2.1.2

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

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Step 2.1.2.1

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

$f' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (- 24 x)$

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 $n = 3$ .

$f' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (- 24 x)$

Step 2.1.2.3

Multiply $3$ by $4$ .

$f' (x) = 12 x^{2} + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (- 24 x)$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

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Step 2.1.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 12 x^{2} - 6 \frac{d}{d x} x^{2} + \frac{d}{d x} (- 24 x)$

Step 2.1.3.2

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

$f' (x) = 12 x^{2} - 6 (2 x) + \frac{d}{d x} (- 24 x)$

Step 2.1.3.3

Multiply $2$ by $- 6$ .

$f' (x) = 12 x^{2} - 12 x + \frac{d}{d x} (- 24 x)$

Step 2.1.4

Evaluate $\frac{d}{d x} [- 24 x]$ .

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Step 2.1.4.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x$ with respect to $x$ is $- 24 \frac{d}{d x} [x]$ .

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

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 $n = 1$ .

$f' (x) = 12 x^{2} - 12 x - 24 \cdot 1$

Step 2.1.4.3

Multiply $- 24$ by $1$ .

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

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $12 x^{2} - 12 x - 24$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 24]$ .

$f'' (x) = \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (- 24)$

Step 2.2.2

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

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Step 2.2.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (- 24)$

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 $n = 2$ .

$f'' (x) = 12 (2 x) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (- 24)$

Step 2.2.2.3

Multiply $2$ by $12$ .

$f'' (x) = 24 x + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (- 24)$

Step 2.2.3

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

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Step 2.2.3.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$f'' (x) = 24 x - 12 \frac{d}{d x} x + \frac{d}{d x} (- 24)$

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 $n = 1$ .

$f'' (x) = 24 x - 12 \cdot 1 + \frac{d}{d x} (- 24)$

Step 2.2.3.3

Multiply $- 12$ by $1$ .

$f'' (x) = 24 x - 12 + \frac{d}{d x} (- 24)$

Step 2.2.4

Differentiate using the Constant Rule.

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Step 2.2.4.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24$ with respect to $x$ is $0$ .

$f'' (x) = 24 x - 12 + 0$

Step 2.2.4.2

Add $24 x - 12$ and $0$ .

$f'' (x) = 24 x - 12$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $24 x - 12$ .

$24 x - 12$

Step 3

Set the second derivative equal to $0$ then solve the equation $24 x - 12 = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$24 x - 12 = 0$

Step 3.2

Add $12$ to both sides of the equation.

$24 x = 12$

Step 3.3

Divide each term in $24 x = 12$ by $24$ and simplify.

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Step 3.3.1

Divide each term in $24 x = 12$ by $24$ .

$\frac{24 x}{24} = \frac{12}{24}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $24$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{24 x}{24} = \frac{12}{24}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{24}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $12$ and $24$ .

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Step 3.3.3.1.1

Factor $12$ out of $12$ .

$x = \frac{12 (1)}{24}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $12$ out of $24$ .

$x = \frac{12 \cdot 1}{12 \cdot 2}$

Step 3.3.3.1.2.2

Cancel the common factor.

$x = \frac{12 \cdot 1}{12 \cdot 2}$

Step 3.3.3.1.2.3

Rewrite the expression.

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

Step 4

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

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Step 4.1

Substitute $\frac{1}{2}$ in $f (x) = 4 x^{3} - 6 x^{2} - 24 x$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

$f (\frac{1}{2}) = 4 {(\frac{1}{2})}^{3} - 6 {(\frac{1}{2})}^{2} - 24 (\frac{1}{2})$

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

Apply the product rule to $\frac{1}{2}$ .

$f (\frac{1}{2}) = 4 (\frac{1^{3}}{2^{3}}) - 6 {(\frac{1}{2})}^{2} - 24 (\frac{1}{2})$

Step 4.1.2.1.2

One to any power is one.

$f (\frac{1}{2}) = 4 (\frac{1}{2^{3}}) - 6 {(\frac{1}{2})}^{2} - 24 (\frac{1}{2})$

Step 4.1.2.1.3

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

$f (\frac{1}{2}) = 4 (\frac{1}{8}) - 6 {(\frac{1}{2})}^{2} - 24 (\frac{1}{2})$

Step 4.1.2.1.4

Cancel the common factor of $4$ .

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Step 4.1.2.1.4.1

Factor $4$ out of $8$ .

$f (\frac{1}{2}) = 4 (\frac{1}{4 (2)}) - 6 {(\frac{1}{2})}^{2} - 24 (\frac{1}{2})$

Step 4.1.2.1.4.2

Cancel the common factor.

$f (\frac{1}{2}) = 4 (\frac{1}{4 \cdot 2}) - 6 {(\frac{1}{2})}^{2} - 24 (\frac{1}{2})$

Step 4.1.2.1.4.3

Rewrite the expression.

$f (\frac{1}{2}) = \frac{1}{2} - 6 {(\frac{1}{2})}^{2} - 24 (\frac{1}{2})$

Step 4.1.2.1.5

Apply the product rule to $\frac{1}{2}$ .

$f (\frac{1}{2}) = \frac{1}{2} - 6 \frac{1^{2}}{2^{2}} - 24 (\frac{1}{2})$

Step 4.1.2.1.6

One to any power is one.

$f (\frac{1}{2}) = \frac{1}{2} - 6 \frac{1}{2^{2}} - 24 (\frac{1}{2})$

Step 4.1.2.1.7

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

$f (\frac{1}{2}) = \frac{1}{2} - 6 (\frac{1}{4}) - 24 (\frac{1}{2})$

Step 4.1.2.1.8

Cancel the common factor of $2$ .

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Step 4.1.2.1.8.1

Factor $2$ out of $- 6$ .

$f (\frac{1}{2}) = \frac{1}{2} + 2 (- 3) (\frac{1}{4}) - 24 (\frac{1}{2})$

Step 4.1.2.1.8.2

Factor $2$ out of $4$ .

$f (\frac{1}{2}) = \frac{1}{2} + 2 \cdot (- 3 \frac{1}{2 \cdot 2}) - 24 (\frac{1}{2})$

Step 4.1.2.1.8.3

Cancel the common factor.

$f (\frac{1}{2}) = \frac{1}{2} + 2 \cdot (- 3 \frac{1}{2 \cdot 2}) - 24 (\frac{1}{2})$

Step 4.1.2.1.8.4

Rewrite the expression.

$f (\frac{1}{2}) = \frac{1}{2} - 3 (\frac{1}{2}) - 24 (\frac{1}{2})$

Step 4.1.2.1.9

Combine $- 3$ and $\frac{1}{2}$ .

$f (\frac{1}{2}) = \frac{1}{2} + \frac{- 3}{2} - 24 (\frac{1}{2})$

Step 4.1.2.1.10

Move the negative in front of the fraction.

$f (\frac{1}{2}) = \frac{1}{2} - \frac{3}{2} - 24 (\frac{1}{2})$

Step 4.1.2.1.11

Cancel the common factor of $2$ .

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Step 4.1.2.1.11.1

Factor $2$ out of $- 24$ .

$f (\frac{1}{2}) = \frac{1}{2} - \frac{3}{2} + 2 (- 12) (\frac{1}{2})$

Step 4.1.2.1.11.2

Cancel the common factor.

$f (\frac{1}{2}) = \frac{1}{2} - \frac{3}{2} + 2 \cdot (- 12 (\frac{1}{2}))$

Step 4.1.2.1.11.3

Rewrite the expression.

$f (\frac{1}{2}) = \frac{1}{2} - \frac{3}{2} - 12$

Step 4.1.2.2

Combine fractions.

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Step 4.1.2.2.1

Combine the numerators over the common denominator.

$f (\frac{1}{2}) = - 12 + \frac{1 - 3}{2}$

Step 4.1.2.2.2

Simplify the expression.

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Step 4.1.2.2.2.1

Subtract $3$ from $1$ .

$f (\frac{1}{2}) = - 12 + \frac{- 2}{2}$

Step 4.1.2.2.2.2

Divide $- 2$ by $2$ .

$f (\frac{1}{2}) = - 12 - 1$

Step 4.1.2.2.2.3

Subtract $1$ from $- 12$ .

$f (\frac{1}{2}) = - 13$

Step 4.1.2.3

The final answer is $- 13$ .

$- 13$

Step 4.2

The point found by substituting $\frac{1}{2}$ in $f (x) = 4 x^{3} - 6 x^{2} - 24 x$ is $(\frac{1}{2}, - 13)$ . This point can be an inflection point.

$(\frac{1}{2}, - 13)$

Step 5

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

$(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, \frac{1}{2})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0.4$ in the expression.

$f'' (0.4) = 24 (0.4) - 12$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $24$ by $0.4$ .

$f'' (0.4) = 9.6 - 12$

Step 6.2.2

Subtract $12$ from $9.6$ .

$f'' (0.4) = - 2.4$

Step 6.2.3

The final answer is $- 2.4$ .

$- 2.4$

Step 6.3

At $0.4$ , the second derivative is $- 2.4$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, \frac{1}{2})$

Decreasing on $(- \infty, \frac{1}{2})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(\frac{1}{2}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.6$ in the expression.

$f'' (0.6) = 24 (0.6) - 12$

Step 7.2

Simplify the result.

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Step 7.2.1

Multiply $24$ by $0.6$ .

$f'' (0.6) = 14.4 - 12$

Step 7.2.2

Subtract $12$ from $14.4$ .

$f'' (0.6) = 2.4$

Step 7.2.3

The final answer is $2.4$ .

$2.4$

Step 7.3

At $0.6$ , the second derivative is $2.4$ . Since this is positive, the second derivative is increasing on the interval $(\frac{1}{2}, \infty)$ .

Increasing on $(\frac{1}{2}, \infty)$ 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 point in this case is $(\frac{1}{2}, - 13)$ .

$(\frac{1}{2}, - 13)$

Step 9