Calculus Examples

$f (x) = 12 x^{3} + 14 x^{2} - 7 x - 9$

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 $12 x^{3} + 14 x^{2} - 7 x - 9$ with respect to $x$ is $\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [14 x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 9]$ .

$\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [14 x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 9]$

Step 1.1.2

Evaluate $\frac{d}{d x} [12 x^{3}]$ .

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Step 1.1.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{3}$ with respect to $x$ is $12 \frac{d}{d x} [x^{3}]$ .

$12 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [14 x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 9]$

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 $n = 3$ .

$12 (3 x^{2}) + \frac{d}{d x} [14 x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 9]$

Step 1.1.2.3

Multiply $3$ by $12$ .

$36 x^{2} + \frac{d}{d x} [14 x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 9]$

Step 1.1.3

Evaluate $\frac{d}{d x} [14 x^{2}]$ .

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Step 1.1.3.1

Since $14$ is constant with respect to $x$ , the derivative of $14 x^{2}$ with respect to $x$ is $14 \frac{d}{d x} [x^{2}]$ .

$36 x^{2} + 14 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 9]$

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 $n = 2$ .

$36 x^{2} + 14 (2 x) + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 9]$

Step 1.1.3.3

Multiply $2$ by $14$ .

$36 x^{2} + 28 x + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 9]$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 7 x]$ .

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Step 1.1.4.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$36 x^{2} + 28 x - 7 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]$

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 $n = 1$ .

$36 x^{2} + 28 x - 7 \cdot 1 + \frac{d}{d x} [- 9]$

Step 1.1.4.3

Multiply $- 7$ by $1$ .

$36 x^{2} + 28 x - 7 + \frac{d}{d x} [- 9]$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$36 x^{2} + 28 x - 7 + 0$

Step 1.1.5.2

Add $36 x^{2} + 28 x - 7$ and $0$ .

$f' (x) = 36 x^{2} + 28 x - 7$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $36 x^{2} + 28 x - 7$ with respect to $x$ is $\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [28 x] + \frac{d}{d x} [- 7]$ .

$\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [28 x] + \frac{d}{d x} [- 7]$

Step 1.2.2

Evaluate $\frac{d}{d x} [36 x^{2}]$ .

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Step 1.2.2.1

Since $36$ is constant with respect to $x$ , the derivative of $36 x^{2}$ with respect to $x$ is $36 \frac{d}{d x} [x^{2}]$ .

$36 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [28 x] + \frac{d}{d x} [- 7]$

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 $n = 2$ .

$36 (2 x) + \frac{d}{d x} [28 x] + \frac{d}{d x} [- 7]$

Step 1.2.2.3

Multiply $2$ by $36$ .

$72 x + \frac{d}{d x} [28 x] + \frac{d}{d x} [- 7]$

Step 1.2.3

Evaluate $\frac{d}{d x} [28 x]$ .

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Step 1.2.3.1

Since $28$ is constant with respect to $x$ , the derivative of $28 x$ with respect to $x$ is $28 \frac{d}{d x} [x]$ .

$72 x + 28 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]$

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 $n = 1$ .

$72 x + 28 \cdot 1 + \frac{d}{d x} [- 7]$

Step 1.2.3.3

Multiply $28$ by $1$ .

$72 x + 28 + \frac{d}{d x} [- 7]$

Step 1.2.4

Differentiate using the Constant Rule.

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Step 1.2.4.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$72 x + 28 + 0$

Step 1.2.4.2

Add $72 x + 28$ and $0$ .

$f'' (x) = 72 x + 28$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $72 x + 28$ .

$72 x + 28$

Step 2

Set the second derivative equal to $0$ then solve the equation $72 x + 28 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$72 x + 28 = 0$

Step 2.2

Subtract $28$ from both sides of the equation.

$72 x = - 28$

Step 2.3

Divide each term in $72 x = - 28$ by $72$ and simplify.

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Step 2.3.1

Divide each term in $72 x = - 28$ by $72$ .

$\frac{72 x}{72} = \frac{- 28}{72}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $72$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{72 x}{72} = \frac{- 28}{72}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 28}{72}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $- 28$ and $72$ .

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Step 2.3.3.1.1

Factor $4$ out of $- 28$ .

$x = \frac{4 (- 7)}{72}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $4$ out of $72$ .

$x = \frac{4 \cdot - 7}{4 \cdot 18}$

Step 2.3.3.1.2.2

Cancel the common factor.

$x = \frac{4 \cdot - 7}{4 \cdot 18}$

Step 2.3.3.1.2.3

Rewrite the expression.

$x = \frac{- 7}{18}$

Step 2.3.3.2

Move the negative in front of the fraction.

$x = - \frac{7}{18}$

Step 3

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

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Step 3.1

Substitute $- \frac{7}{18}$ in $f (x) = 12 x^{3} + 14 x^{2} - 7 x - 9$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $- \frac{7}{18}$ in the expression.

$f (- \frac{7}{18}) = 12 {(- \frac{7}{18})}^{3} + 14 {(- \frac{7}{18})}^{2} - 7 (- \frac{7}{18}) - 9$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.2.1.1.1

Apply the product rule to $- \frac{7}{18}$ .

$f (- \frac{7}{18}) = 12 ({(- 1)}^{3} {(\frac{7}{18})}^{3}) + 14 {(- \frac{7}{18})}^{2} - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.1.2

Apply the product rule to $\frac{7}{18}$ .

$f (- \frac{7}{18}) = 12 ({(- 1)}^{3} (\frac{7^{3}}{18^{3}})) + 14 {(- \frac{7}{18})}^{2} - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.2

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

$f (- \frac{7}{18}) = 12 (- \frac{7^{3}}{18^{3}}) + 14 {(- \frac{7}{18})}^{2} - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.3

Raise $7$ to the power of $3$ .

$f (- \frac{7}{18}) = 12 (- \frac{343}{18^{3}}) + 14 {(- \frac{7}{18})}^{2} - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.4

Raise $18$ to the power of $3$ .

$f (- \frac{7}{18}) = 12 (- \frac{343}{5832}) + 14 {(- \frac{7}{18})}^{2} - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.5

Cancel the common factor of $12$ .

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Step 3.1.2.1.5.1

Move the leading negative in $- \frac{343}{5832}$ into the numerator.

$f (- \frac{7}{18}) = 12 (\frac{- 343}{5832}) + 14 {(- \frac{7}{18})}^{2} - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.5.2

Factor $12$ out of $5832$ .

$f (- \frac{7}{18}) = 12 (\frac{- 343}{12 (486)}) + 14 {(- \frac{7}{18})}^{2} - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.5.3

Cancel the common factor.

$f (- \frac{7}{18}) = 12 (\frac{- 343}{12 \cdot 486}) + 14 {(- \frac{7}{18})}^{2} - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.5.4

Rewrite the expression.

$f (- \frac{7}{18}) = \frac{- 343}{486} + 14 {(- \frac{7}{18})}^{2} - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.6

Move the negative in front of the fraction.

$f (- \frac{7}{18}) = - \frac{343}{486} + 14 {(- \frac{7}{18})}^{2} - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.2.1.7.1

Apply the product rule to $- \frac{7}{18}$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + 14 ({(- 1)}^{2} {(\frac{7}{18})}^{2}) - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.7.2

Apply the product rule to $\frac{7}{18}$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + 14 ({(- 1)}^{2} (\frac{7^{2}}{18^{2}})) - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.8

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

$f (- \frac{7}{18}) = - \frac{343}{486} + 14 (1 (\frac{7^{2}}{18^{2}})) - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.9

Multiply $\frac{7^{2}}{18^{2}}$ by $1$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + 14 (\frac{7^{2}}{18^{2}}) - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.10

Raise $7$ to the power of $2$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + 14 (\frac{49}{18^{2}}) - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.11

Raise $18$ to the power of $2$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + 14 (\frac{49}{324}) - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.12

Cancel the common factor of $2$ .

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Step 3.1.2.1.12.1

Factor $2$ out of $14$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + 2 (7) (\frac{49}{324}) - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.12.2

Factor $2$ out of $324$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + 2 \cdot (7 (\frac{49}{2 \cdot 162})) - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.12.3

Cancel the common factor.

$f (- \frac{7}{18}) = - \frac{343}{486} + 2 \cdot (7 (\frac{49}{2 \cdot 162})) - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.12.4

Rewrite the expression.

$f (- \frac{7}{18}) = - \frac{343}{486} + 7 (\frac{49}{162}) - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.13

Combine $7$ and $\frac{49}{162}$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{7 \cdot 49}{162} - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.14

Multiply $7$ by $49$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343}{162} - 7 (- \frac{7}{18}) - 9$

Step 3.1.2.1.15

Multiply $- 7 (- \frac{7}{18})$ .

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Step 3.1.2.1.15.1

Multiply $- 1$ by $- 7$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343}{162} + 7 (\frac{7}{18}) - 9$

Step 3.1.2.1.15.2

Combine $7$ and $\frac{7}{18}$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343}{162} + \frac{7 \cdot 7}{18} - 9$

Step 3.1.2.1.15.3

Multiply $7$ by $7$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343}{162} + \frac{49}{18} - 9$

Step 3.1.2.2

Find the common denominator.

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Step 3.1.2.2.1

Multiply $\frac{343}{162}$ by $\frac{3}{3}$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343}{162} \cdot \frac{3}{3} + \frac{49}{18} - 9$

Step 3.1.2.2.2

Multiply $\frac{343}{162}$ by $\frac{3}{3}$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343 \cdot 3}{162 \cdot 3} + \frac{49}{18} - 9$

Step 3.1.2.2.3

Multiply $\frac{49}{18}$ by $\frac{27}{27}$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343 \cdot 3}{162 \cdot 3} + \frac{49}{18} \cdot \frac{27}{27} - 9$

Step 3.1.2.2.4

Multiply $\frac{49}{18}$ by $\frac{27}{27}$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343 \cdot 3}{162 \cdot 3} + \frac{49 \cdot 27}{18 \cdot 27} - 9$

Step 3.1.2.2.5

Write $- 9$ as a fraction with denominator $1$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343 \cdot 3}{162 \cdot 3} + \frac{49 \cdot 27}{18 \cdot 27} + \frac{- 9}{1}$

Step 3.1.2.2.6

Multiply $\frac{- 9}{1}$ by $\frac{486}{486}$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343 \cdot 3}{162 \cdot 3} + \frac{49 \cdot 27}{18 \cdot 27} + \frac{- 9}{1} \cdot \frac{486}{486}$

Step 3.1.2.2.7

Multiply $\frac{- 9}{1}$ by $\frac{486}{486}$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343 \cdot 3}{162 \cdot 3} + \frac{49 \cdot 27}{18 \cdot 27} + \frac{- 9 \cdot 486}{486}$

Step 3.1.2.2.8

Reorder the factors of $162 \cdot 3$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343 \cdot 3}{3 \cdot 162} + \frac{49 \cdot 27}{18 \cdot 27} + \frac{- 9 \cdot 486}{486}$

Step 3.1.2.2.9

Multiply $3$ by $162$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343 \cdot 3}{486} + \frac{49 \cdot 27}{18 \cdot 27} + \frac{- 9 \cdot 486}{486}$

Step 3.1.2.2.10

Multiply $18$ by $27$ .

$f (- \frac{7}{18}) = - \frac{343}{486} + \frac{343 \cdot 3}{486} + \frac{49 \cdot 27}{486} + \frac{- 9 \cdot 486}{486}$

Step 3.1.2.3

Combine the numerators over the common denominator.

$f (- \frac{7}{18}) = \frac{- 343 + 343 \cdot 3 + 49 \cdot 27 - 9 \cdot 486}{486}$

Step 3.1.2.4

Simplify each term.

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Step 3.1.2.4.1

Multiply $343$ by $3$ .

$f (- \frac{7}{18}) = \frac{- 343 + 1029 + 49 \cdot 27 - 9 \cdot 486}{486}$

Step 3.1.2.4.2

Multiply $49$ by $27$ .

$f (- \frac{7}{18}) = \frac{- 343 + 1029 + 1323 - 9 \cdot 486}{486}$

Step 3.1.2.4.3

Multiply $- 9$ by $486$ .

$f (- \frac{7}{18}) = \frac{- 343 + 1029 + 1323 - 4374}{486}$

Step 3.1.2.5

Simplify the expression.

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Step 3.1.2.5.1

Add $- 343$ and $1029$ .

$f (- \frac{7}{18}) = \frac{686 + 1323 - 4374}{486}$

Step 3.1.2.5.2

Add $686$ and $1323$ .

$f (- \frac{7}{18}) = \frac{2009 - 4374}{486}$

Step 3.1.2.5.3

Subtract $4374$ from $2009$ .

$f (- \frac{7}{18}) = \frac{- 2365}{486}$

Step 3.1.2.5.4

Move the negative in front of the fraction.

$f (- \frac{7}{18}) = - \frac{2365}{486}$

Step 3.1.2.6

The final answer is $- \frac{2365}{486}$ .

$- \frac{2365}{486}$

Step 3.2

The point found by substituting $- \frac{7}{18}$ in $f (x) = 12 x^{3} + 14 x^{2} - 7 x - 9$ is $(- \frac{7}{18}, - \frac{2365}{486})$ . This point can be an inflection point.

$(- \frac{7}{18}, - \frac{2365}{486})$

Step 4

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

$(- \infty, - \frac{7}{18}) \cup (- \frac{7}{18}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \frac{7}{18})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 0.4\overline{8}$ in the expression.

$f'' (- 0.4\overline{8}) = 72 (- 0.4\overline{8}) + 28$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $72$ by $- 0.4\overline{8}$ .

$f'' (- 0.4\overline{8}) = - 35.2 + 28$

Step 5.2.2

Add $- 35.2$ and $28$ .

$f'' (- 0.4\overline{8}) = - 7.2$

Step 5.2.3

The final answer is $- 7.2$ .

$- 7.2$

Step 5.3

At $- 0.4\overline{8}$ , the second derivative is $- 7.2$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - \frac{7}{18})$

Decreasing on $(- \infty, - \frac{7}{18})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- \frac{7}{18}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.2\overline{8}$ in the expression.

$f'' (- 0.2\overline{8}) = 72 (- 0.2\overline{8}) + 28$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $72$ by $- 0.2\overline{8}$ .

$f'' (- 0.2\overline{8}) = - 20.8 + 28$

Step 6.2.2

Add $- 20.8$ and $28$ .

$f'' (- 0.2\overline{8}) = 7.1\overline{9}$

Step 6.2.3

The final answer is $7.1\overline{9}$ .

$7.1\overline{9}$

Step 6.3

At $- 0.2\overline{8}$ , the second derivative is $7.1\overline{9}$ . Since this is positive, the second derivative is increasing on the interval $(- \frac{7}{18}, \infty)$ .

Increasing on $(- \frac{7}{18}, \infty)$ 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 $(- \frac{7}{18}, - \frac{2365}{486})$ .

$(- \frac{7}{18}, - \frac{2365}{486})$

Step 8