Calculus Examples

$f (x) = - 3 x^{\frac{4}{3}} + 12 x - 10$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $- 3 x^{\frac{4}{3}} + 12 x - 10$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{\frac{4}{3}}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$ .

$\frac{d}{d x} [- 3 x^{\frac{4}{3}}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2

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

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Step 1.2.1

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

$- 3 \frac{d}{d x} [x^{\frac{4}{3}}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

$- 3 (\frac{4}{3} x^{\frac{4}{3} - 1}) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 3 (\frac{4}{3} x^{\frac{4}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.4

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

$- 3 (\frac{4}{3} x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.5

Combine the numerators over the common denominator.

$- 3 (\frac{4}{3} x^{\frac{4 - 1 \cdot 3}{3}}) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{4}{3} x^{\frac{4 - 3}{3}}) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.6.2

Subtract $3$ from $4$ .

$- 3 (\frac{4}{3} x^{\frac{1}{3}}) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.7

Combine $\frac{4}{3}$ and $x^{\frac{1}{3}}$ .

$- 3 \frac{4 x^{\frac{1}{3}}}{3} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.8

Combine $- 3$ and $\frac{4 x^{\frac{1}{3}}}{3}$ .

$\frac{- 3 (4 x^{\frac{1}{3}})}{3} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.9

Multiply $4$ by $- 3$ .

$\frac{- 12 x^{\frac{1}{3}}}{3} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.10

Factor $3$ out of $- 12 x^{\frac{1}{3}}$ .

$\frac{3 (- 4 x^{\frac{1}{3}})}{3} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.11

Cancel the common factors.

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Step 1.2.11.1

Factor $3$ out of $3$ .

$\frac{3 (- 4 x^{\frac{1}{3}})}{3 (1)} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.11.2

Cancel the common factor.

$\frac{3 (- 4 x^{\frac{1}{3}})}{3 \cdot 1} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.11.3

Rewrite the expression.

$\frac{- 4 x^{\frac{1}{3}}}{1} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.2.11.4

Divide $- 4 x^{\frac{1}{3}}$ by $1$ .

$- 4 x^{\frac{1}{3}} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 1.3

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

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Step 1.3.1

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

$- 4 x^{\frac{1}{3}} + 12 \frac{d}{d x} [x] + \frac{d}{d x} [- 10]$

Step 1.3.2

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

$- 4 x^{\frac{1}{3}} + 12 \cdot 1 + \frac{d}{d x} [- 10]$

Step 1.3.3

Multiply $12$ by $1$ .

$- 4 x^{\frac{1}{3}} + 12 + \frac{d}{d x} [- 10]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10$ with respect to $x$ is $0$ .

$- 4 x^{\frac{1}{3}} + 12 + 0$

Step 1.4.2

Add $- 4 x^{\frac{1}{3}} + 12$ and $0$ .

$- 4 x^{\frac{1}{3}} + 12$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- 4 x^{\frac{1}{3}} + 12$ with respect to $x$ is $\frac{d}{d x} [- 4 x^{\frac{1}{3}}] + \frac{d}{d x} [12]$ .

$f'' (x) = \frac{d}{d x} (- 4 x^{\frac{1}{3}}) + \frac{d}{d x} (12)$

Step 2.2

Evaluate $\frac{d}{d x} [- 4 x^{\frac{1}{3}}]$ .

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Step 2.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{\frac{1}{3}}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$f'' (x) = - 4 \frac{d}{d x} x^{\frac{1}{3}} + \frac{d}{d x} (12)$

Step 2.2.2

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

$f'' (x) = - 4 (\frac{1}{3} x^{\frac{1}{3} - 1}) + \frac{d}{d x} (12)$

Step 2.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f'' (x) = - 4 (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} (12)$

Step 2.2.4

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

$f'' (x) = - 4 (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} (12)$

Step 2.2.5

Combine the numerators over the common denominator.

$f'' (x) = - 4 (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}}) + \frac{d}{d x} (12)$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $3$ .

$f'' (x) = - 4 (\frac{1}{3} x^{\frac{1 - 3}{3}}) + \frac{d}{d x} (12)$

Step 2.2.6.2

Subtract $3$ from $1$ .

$f'' (x) = - 4 (\frac{1}{3} x^{\frac{- 2}{3}}) + \frac{d}{d x} (12)$

Step 2.2.7

Move the negative in front of the fraction.

$f'' (x) = - 4 (\frac{1}{3} x^{- \frac{2}{3}}) + \frac{d}{d x} (12)$

Step 2.2.8

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

$f'' (x) = - 4 \frac{x^{- \frac{2}{3}}}{3} + \frac{d}{d x} (12)$

Step 2.2.9

Combine $- 4$ and $\frac{x^{- \frac{2}{3}}}{3}$ .

$f'' (x) = \frac{- 4 x^{- \frac{2}{3}}}{3} + \frac{d}{d x} (12)$

Step 2.2.10

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{- 4}{3 x^{\frac{2}{3}}} + \frac{d}{d x} (12)$

Step 2.2.11

Move the negative in front of the fraction.

$f'' (x) = - \frac{4}{3 x^{\frac{2}{3}}} + \frac{d}{d x} (12)$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

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

$f'' (x) = - \frac{4}{3 x^{\frac{2}{3}}} + 0$

Step 2.3.2

Add $- \frac{4}{3 x^{\frac{2}{3}}}$ and $0$ .

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

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- 4 x^{\frac{1}{3}} + 12 = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

By the Sum Rule, the derivative of $- 3 x^{\frac{4}{3}} + 12 x - 10$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{\frac{4}{3}}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$ .

$\frac{d}{d x} [- 3 x^{\frac{4}{3}}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2

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

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Step 4.1.2.1

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

$- 3 \frac{d}{d x} [x^{\frac{4}{3}}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

$- 3 (\frac{4}{3} x^{\frac{4}{3} - 1}) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 3 (\frac{4}{3} x^{\frac{4}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.4

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

$- 3 (\frac{4}{3} x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.5

Combine the numerators over the common denominator.

$- 3 (\frac{4}{3} x^{\frac{4 - 1 \cdot 3}{3}}) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{4}{3} x^{\frac{4 - 3}{3}}) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.6.2

Subtract $3$ from $4$ .

$- 3 (\frac{4}{3} x^{\frac{1}{3}}) + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.7

Combine $\frac{4}{3}$ and $x^{\frac{1}{3}}$ .

$- 3 \frac{4 x^{\frac{1}{3}}}{3} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.8

Combine $- 3$ and $\frac{4 x^{\frac{1}{3}}}{3}$ .

$\frac{- 3 (4 x^{\frac{1}{3}})}{3} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.9

Multiply $4$ by $- 3$ .

$\frac{- 12 x^{\frac{1}{3}}}{3} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.10

Factor $3$ out of $- 12 x^{\frac{1}{3}}$ .

$\frac{3 (- 4 x^{\frac{1}{3}})}{3} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.11

Cancel the common factors.

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Step 4.1.2.11.1

Factor $3$ out of $3$ .

$\frac{3 (- 4 x^{\frac{1}{3}})}{3 (1)} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.11.2

Cancel the common factor.

$\frac{3 (- 4 x^{\frac{1}{3}})}{3 \cdot 1} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.11.3

Rewrite the expression.

$\frac{- 4 x^{\frac{1}{3}}}{1} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.2.11.4

Divide $- 4 x^{\frac{1}{3}}$ by $1$ .

$- 4 x^{\frac{1}{3}} + \frac{d}{d x} [12 x] + \frac{d}{d x} [- 10]$

Step 4.1.3

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

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Step 4.1.3.1

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

$- 4 x^{\frac{1}{3}} + 12 \frac{d}{d x} [x] + \frac{d}{d x} [- 10]$

Step 4.1.3.2

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

$- 4 x^{\frac{1}{3}} + 12 \cdot 1 + \frac{d}{d x} [- 10]$

Step 4.1.3.3

Multiply $12$ by $1$ .

$- 4 x^{\frac{1}{3}} + 12 + \frac{d}{d x} [- 10]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10$ with respect to $x$ is $0$ .

$- 4 x^{\frac{1}{3}} + 12 + 0$

Step 4.1.4.2

Add $- 4 x^{\frac{1}{3}} + 12$ and $0$ .

$f' (x) = - 4 x^{\frac{1}{3}} + 12$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 4 x^{\frac{1}{3}} + 12$ .

$- 4 x^{\frac{1}{3}} + 12$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 4 x^{\frac{1}{3}} + 12 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- 4 x^{\frac{1}{3}} + 12 = 0$

Step 5.2

Subtract $12$ from both sides of the equation.

$- 4 x^{\frac{1}{3}} = - 12$

Step 5.3

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(- 4 x^{\frac{1}{3}})}^{3} = {(- 12)}^{3}$

Step 5.4

Simplify the exponent.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Simplify ${(- 4 x^{\frac{1}{3}})}^{3}$ .

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Step 5.4.1.1.1

Apply the product rule to $- 4 x^{\frac{1}{3}}$ .

${(- 4)}^{3} {(x^{\frac{1}{3}})}^{3} = {(- 12)}^{3}$

Step 5.4.1.1.2

Raise $- 4$ to the power of $3$ .

$- 64 {(x^{\frac{1}{3}})}^{3} = {(- 12)}^{3}$

Step 5.4.1.1.3

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 5.4.1.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 64 x^{\frac{1}{3} \cdot 3} = {(- 12)}^{3}$

Step 5.4.1.1.3.2

Cancel the common factor of $3$ .

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Step 5.4.1.1.3.2.1

Cancel the common factor.

$- 64 x^{\frac{1}{3} \cdot 3} = {(- 12)}^{3}$

Step 5.4.1.1.3.2.2

Rewrite the expression.

$- 64 x^{1} = {(- 12)}^{3}$

Step 5.4.1.1.4

Simplify.

$- 64 x = {(- 12)}^{3}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Raise $- 12$ to the power of $3$ .

$- 64 x = - 1728$

Step 5.5

Divide each term in $- 64 x = - 1728$ by $- 64$ and simplify.

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Step 5.5.1

Divide each term in $- 64 x = - 1728$ by $- 64$ .

$\frac{- 64 x}{- 64} = \frac{- 1728}{- 64}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $- 64$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{- 64 x}{- 64} = \frac{- 1728}{- 64}$

Step 5.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1728}{- 64}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Divide $- 1728$ by $- 64$ .

$x = 27$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 27$

Step 8

Evaluate the second derivative at $x = 27$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{4}{3 {(27)}^{\frac{2}{3}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

Rewrite $27$ as $3^{3}$ .

$- \frac{4}{3 \cdot {(3^{3})}^{\frac{2}{3}}}$

Step 9.1.2

Multiply the exponents in ${(3^{3})}^{\frac{2}{3}}$ .

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Step 9.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{4}{3 \cdot 3^{3 (\frac{2}{3})}}$

Step 9.1.2.2

Cancel the common factor of $3$ .

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Step 9.1.2.2.1

Cancel the common factor.

$- \frac{4}{3 \cdot 3^{3 (\frac{2}{3})}}$

Step 9.1.2.2.2

Rewrite the expression.

$- \frac{4}{3 \cdot 3^{2}}$

Step 9.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{4}{3^{1 + 2}}$

Step 9.1.4

Add $1$ and $2$ .

$- \frac{4}{3^{3}}$

Step 9.2

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

$- \frac{4}{27}$

Step 10

$x = 27$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 27$ is a local maximum

Step 11

Find the y-value when $x = 27$ .

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Step 11.1

Replace the variable $x$ with $27$ in the expression.

$f (27) = - 3 {(27)}^{\frac{4}{3}} + 12 (27) - 10$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Rewrite $27$ as $3^{3}$ .

$f (27) = - 3 {(3^{3})}^{\frac{4}{3}} + 12 (27) - 10$

Step 11.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (27) = - 3 \cdot 3^{3 (\frac{4}{3})} + 12 (27) - 10$

Step 11.2.1.3

Cancel the common factor of $3$ .

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Step 11.2.1.3.1

Cancel the common factor.

$f (27) = - 3 \cdot 3^{3 (\frac{4}{3})} + 12 (27) - 10$

Step 11.2.1.3.2

Rewrite the expression.

$f (27) = - 3 \cdot 3^{4} + 12 (27) - 10$

Step 11.2.1.4

Raise $3$ to the power of $4$ .

$f (27) = - 3 \cdot 81 + 12 (27) - 10$

Step 11.2.1.5

Multiply $- 3$ by $81$ .

$f (27) = - 243 + 12 (27) - 10$

Step 11.2.1.6

Multiply $12$ by $27$ .

$f (27) = - 243 + 324 - 10$

Step 11.2.2

Simplify by adding and subtracting.

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Step 11.2.2.1

Add $- 243$ and $324$ .

$f (27) = 81 - 10$

Step 11.2.2.2

Subtract $10$ from $81$ .

$f (27) = 71$

Step 11.2.3

The final answer is $71$ .

$y = 71$

Step 12

These are the local extrema for $f (x) = - 3 x^{\frac{4}{3}} + 12 x - 10$ .

$(27, 71)$ is a local maxima

Step 13