Calculus Examples

$f (x) = 2 x^{3} - 9 x^{2} + 12 x - 6$

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 $2 x^{3} - 9 x^{2} + 12 x - 6$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 6]$

Step 1.1.2

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

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Step 1.1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 6]$

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 $n = 3$ .

$2 (3 x^{2}) + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 6]$

Step 1.1.2.3

Multiply $3$ by $2$ .

$6 x^{2} + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 6]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 9 x^{2}]$ .

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Step 1.1.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{2}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{2}]$ .

$6 x^{2} - 9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 6]$

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 $n = 2$ .

$6 x^{2} - 9 (2 x) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 6]$

Step 1.1.3.3

Multiply $2$ by $- 9$ .

$6 x^{2} - 18 x + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 6]$

Step 1.1.4

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

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Step 1.1.4.1

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

$6 x^{2} - 18 x + 12 \frac{d}{d x} [x] + \frac{d}{d x} [- 6]$

Step 1.1.4.2

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

$6 x^{2} - 18 x + 12 \cdot 1 + \frac{d}{d x} [- 6]$

Step 1.1.4.3

Multiply $12$ by $1$ .

$6 x^{2} - 18 x + 12 + \frac{d}{d x} [- 6]$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$6 x^{2} - 18 x + 12 + 0$

Step 1.1.5.2

Add $6 x^{2} - 18 x + 12$ and $0$ .

$f' (x) = 6 x^{2} - 18 x + 12$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $6 x^{2} - 18 x + 12$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$ .

$\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$

Step 1.2.2

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

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Step 1.2.2.1

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

$6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$

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 $n = 2$ .

$6 (2 x) + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$

Step 1.2.2.3

Multiply $2$ by $6$ .

$12 x + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [12]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 18 x]$ .

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Step 1.2.3.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x$ with respect to $x$ is $- 18 \frac{d}{d x} [x]$ .

$12 x - 18 \frac{d}{d x} [x] + \frac{d}{d x} [12]$

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 $n = 1$ .

$12 x - 18 \cdot 1 + \frac{d}{d x} [12]$

Step 1.2.3.3

Multiply $- 18$ by $1$ .

$12 x - 18 + \frac{d}{d x} [12]$

Step 1.2.4

Differentiate using the Constant Rule.

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Step 1.2.4.1

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$12 x - 18 + 0$

Step 1.2.4.2

Add $12 x - 18$ and $0$ .

$f'' (x) = 12 x - 18$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $12 x - 18$ .

$12 x - 18$

Step 2

Set the second derivative equal to $0$ then solve the equation $12 x - 18 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$12 x - 18 = 0$

Step 2.2

Add $18$ to both sides of the equation.

$12 x = 18$

Step 2.3

Divide each term in $12 x = 18$ by $12$ and simplify.

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Step 2.3.1

Divide each term in $12 x = 18$ by $12$ .

$\frac{12 x}{12} = \frac{18}{12}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $12$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{18}{12}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{12}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $18$ and $12$ .

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Step 2.3.3.1.1

Factor $6$ out of $18$ .

$x = \frac{6 (3)}{12}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $6$ out of $12$ .

$x = \frac{6 \cdot 3}{6 \cdot 2}$

Step 2.3.3.1.2.2

Cancel the common factor.

$x = \frac{6 \cdot 3}{6 \cdot 2}$

Step 2.3.3.1.2.3

Rewrite the expression.

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

Step 3

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

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Step 3.1

Substitute $\frac{3}{2}$ in $f (x) = 2 x^{3} - 9 x^{2} + 12 x - 6$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $\frac{3}{2}$ in the expression.

$f (\frac{3}{2}) = 2 {(\frac{3}{2})}^{3} - 9 {(\frac{3}{2})}^{2} + 12 (\frac{3}{2}) - 6$

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

Apply the product rule to $\frac{3}{2}$ .

$f (\frac{3}{2}) = 2 (\frac{3^{3}}{2^{3}}) - 9 {(\frac{3}{2})}^{2} + 12 (\frac{3}{2}) - 6$

Step 3.1.2.1.2

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

$f (\frac{3}{2}) = 2 (\frac{27}{2^{3}}) - 9 {(\frac{3}{2})}^{2} + 12 (\frac{3}{2}) - 6$

Step 3.1.2.1.3

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

$f (\frac{3}{2}) = 2 (\frac{27}{8}) - 9 {(\frac{3}{2})}^{2} + 12 (\frac{3}{2}) - 6$

Step 3.1.2.1.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.4.1

Factor $2$ out of $8$ .

$f (\frac{3}{2}) = 2 (\frac{27}{2 (4)}) - 9 {(\frac{3}{2})}^{2} + 12 (\frac{3}{2}) - 6$

Step 3.1.2.1.4.2

Cancel the common factor.

$f (\frac{3}{2}) = 2 (\frac{27}{2 \cdot 4}) - 9 {(\frac{3}{2})}^{2} + 12 (\frac{3}{2}) - 6$

Step 3.1.2.1.4.3

Rewrite the expression.

$f (\frac{3}{2}) = \frac{27}{4} - 9 {(\frac{3}{2})}^{2} + 12 (\frac{3}{2}) - 6$

Step 3.1.2.1.5

Apply the product rule to $\frac{3}{2}$ .

$f (\frac{3}{2}) = \frac{27}{4} - 9 \frac{3^{2}}{2^{2}} + 12 (\frac{3}{2}) - 6$

Step 3.1.2.1.6

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

$f (\frac{3}{2}) = \frac{27}{4} - 9 \frac{9}{2^{2}} + 12 (\frac{3}{2}) - 6$

Step 3.1.2.1.7

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

$f (\frac{3}{2}) = \frac{27}{4} - 9 (\frac{9}{4}) + 12 (\frac{3}{2}) - 6$

Step 3.1.2.1.8

Multiply $- 9 (\frac{9}{4})$ .

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Step 3.1.2.1.8.1

Combine $- 9$ and $\frac{9}{4}$ .

$f (\frac{3}{2}) = \frac{27}{4} + \frac{- 9 \cdot 9}{4} + 12 (\frac{3}{2}) - 6$

Step 3.1.2.1.8.2

Multiply $- 9$ by $9$ .

$f (\frac{3}{2}) = \frac{27}{4} + \frac{- 81}{4} + 12 (\frac{3}{2}) - 6$

Step 3.1.2.1.9

Move the negative in front of the fraction.

$f (\frac{3}{2}) = \frac{27}{4} - \frac{81}{4} + 12 (\frac{3}{2}) - 6$

Step 3.1.2.1.10

Cancel the common factor of $2$ .

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Step 3.1.2.1.10.1

Factor $2$ out of $12$ .

$f (\frac{3}{2}) = \frac{27}{4} - \frac{81}{4} + 2 (6) (\frac{3}{2}) - 6$

Step 3.1.2.1.10.2

Cancel the common factor.

$f (\frac{3}{2}) = \frac{27}{4} - \frac{81}{4} + 2 \cdot (6 (\frac{3}{2})) - 6$

Step 3.1.2.1.10.3

Rewrite the expression.

$f (\frac{3}{2}) = \frac{27}{4} - \frac{81}{4} + 6 \cdot 3 - 6$

Step 3.1.2.1.11

Multiply $6$ by $3$ .

$f (\frac{3}{2}) = \frac{27}{4} - \frac{81}{4} + 18 - 6$

Step 3.1.2.2

Combine fractions.

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Step 3.1.2.2.1

Combine the numerators over the common denominator.

$f (\frac{3}{2}) = 18 - 6 + \frac{27 - 81}{4}$

Step 3.1.2.2.2

Subtract $81$ from $27$ .

$f (\frac{3}{2}) = 18 - 6 + \frac{- 54}{4}$

Step 3.1.2.3

Find the common denominator.

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Step 3.1.2.3.1

Write $18$ as a fraction with denominator $1$ .

$f (\frac{3}{2}) = \frac{18}{1} - 6 + \frac{- 54}{4}$

Step 3.1.2.3.2

Multiply $\frac{18}{1}$ by $\frac{4}{4}$ .

$f (\frac{3}{2}) = \frac{18}{1} \cdot \frac{4}{4} - 6 + \frac{- 54}{4}$

Step 3.1.2.3.3

Multiply $\frac{18}{1}$ by $\frac{4}{4}$ .

$f (\frac{3}{2}) = \frac{18 \cdot 4}{4} - 6 + \frac{- 54}{4}$

Step 3.1.2.3.4

Write $- 6$ as a fraction with denominator $1$ .

$f (\frac{3}{2}) = \frac{18 \cdot 4}{4} + \frac{- 6}{1} + \frac{- 54}{4}$

Step 3.1.2.3.5

Multiply $\frac{- 6}{1}$ by $\frac{4}{4}$ .

$f (\frac{3}{2}) = \frac{18 \cdot 4}{4} + \frac{- 6}{1} \cdot \frac{4}{4} + \frac{- 54}{4}$

Step 3.1.2.3.6

Multiply $\frac{- 6}{1}$ by $\frac{4}{4}$ .

$f (\frac{3}{2}) = \frac{18 \cdot 4}{4} + \frac{- 6 \cdot 4}{4} + \frac{- 54}{4}$

Step 3.1.2.4

Combine the numerators over the common denominator.

$f (\frac{3}{2}) = \frac{18 \cdot 4 - 6 \cdot 4 - 54}{4}$

Step 3.1.2.5

Simplify each term.

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Step 3.1.2.5.1

Multiply $18$ by $4$ .

$f (\frac{3}{2}) = \frac{72 - 6 \cdot 4 - 54}{4}$

Step 3.1.2.5.2

Multiply $- 6$ by $4$ .

$f (\frac{3}{2}) = \frac{72 - 24 - 54}{4}$

Step 3.1.2.6

Reduce the expression by cancelling the common factors.

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Step 3.1.2.6.1

Subtract $24$ from $72$ .

$f (\frac{3}{2}) = \frac{48 - 54}{4}$

Step 3.1.2.6.2

Subtract $54$ from $48$ .

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

Step 3.1.2.6.3

Cancel the common factor of $- 6$ and $4$ .

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Step 3.1.2.6.3.1

Factor $2$ out of $- 6$ .

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

Step 3.1.2.6.3.2

Cancel the common factors.

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Step 3.1.2.6.3.2.1

Factor $2$ out of $4$ .

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

Step 3.1.2.6.3.2.2

Cancel the common factor.

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

Step 3.1.2.6.3.2.3

Rewrite the expression.

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

Step 3.1.2.6.4

Move the negative in front of the fraction.

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

Step 3.1.2.7

The final answer is $- \frac{3}{2}$ .

$- \frac{3}{2}$

Step 3.2

The point found by substituting $\frac{3}{2}$ in $f (x) = 2 x^{3} - 9 x^{2} + 12 x - 6$ is $(\frac{3}{2}, - \frac{3}{2})$ . This point can be an inflection point.

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

Step 4

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

$(- \infty, \frac{3}{2}) \cup (\frac{3}{2}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{3}{2})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $1.4$ in the expression.

$f'' (1.4) = 12 (1.4) - 18$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $12$ by $1.4$ .

$f'' (1.4) = 16.8 - 18$

Step 5.2.2

Subtract $18$ from $16.8$ .

$f'' (1.4) = - 1.2$

Step 5.2.3

The final answer is $- 1.2$ .

$- 1.2$

Step 5.3

At $1.4$ , the second derivative is $- 1.2$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, \frac{3}{2})$

Decreasing on $(- \infty, \frac{3}{2})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(\frac{3}{2}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $1.6$ in the expression.

$f'' (1.6) = 12 (1.6) - 18$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $12$ by $1.6$ .

$f'' (1.6) = 19.2 - 18$

Step 6.2.2

Subtract $18$ from $19.2$ .

$f'' (1.6) = 1.2$

Step 6.2.3

The final answer is $1.2$ .

$1.2$

Step 6.3

At $1.6$ , the second derivative is $1.2$ . Since this is positive, the second derivative is increasing on the interval $(\frac{3}{2}, \infty)$ .

Increasing on $(\frac{3}{2}, \infty)$ since $f'' (x) > 0$

Step 7

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

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

Step 8