Calculus Examples

$f (x) = x (x - 9 \sqrt{x})$

Step 1

Find the second derivative.

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Find the first derivative.

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$f' (x) = \frac{d}{d x} (x (x - 9 x^{\frac{1}{2}}))$

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x - 9 x^{\frac{1}{2}}$ .

$f' (x) = x \frac{d}{d x} (x - 9 x^{\frac{1}{2}}) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

Differentiate.

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By the Sum Rule, the derivative of $x - 9 x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 9 x^{\frac{1}{2}}]$ .

$f' (x) = x (\frac{d}{d x} (x) + \frac{d}{d x} (- 9 x^{\frac{1}{2}})) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

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

$f' (x) = x (1 + \frac{d}{d x} (- 9 x^{\frac{1}{2}})) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{\frac{1}{2}}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$f' (x) = x (1 - 9 \frac{d}{d x} x^{\frac{1}{2}}) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

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

$f' (x) = x (1 - 9 (\frac{1}{2} x^{\frac{1}{2} - 1})) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (x) = x (1 - 9 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

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

$f' (x) = x (1 - 9 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

Combine the numerators over the common denominator.

$f' (x) = x (1 - 9 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

Simplify the numerator.

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Multiply $- 1$ by $2$ .

$f' (x) = x (1 - 9 (\frac{1}{2} x^{\frac{1 - 2}{2}})) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

Subtract $2$ from $1$ .

$f' (x) = x (1 - 9 (\frac{1}{2} x^{\frac{- 1}{2}})) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

Combine fractions.

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Move the negative in front of the fraction.

$f' (x) = x (1 - 9 (\frac{1}{2} x^{- \frac{1}{2}})) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

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

$f' (x) = x (1 - 9 \frac{x^{- \frac{1}{2}}}{2}) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

Combine $- 9$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$f' (x) = x (1 + \frac{- 9 x^{- \frac{1}{2}}}{2}) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

Simplify the expression.

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

$f' (x) = x (1 + \frac{- 9}{2 x^{\frac{1}{2}}}) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

Move the negative in front of the fraction.

$f' (x) = x (1 - \frac{9}{2 x^{\frac{1}{2}}}) + (x - 9 x^{\frac{1}{2}}) \frac{d}{d x} (x)$

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

$f' (x) = x (1 - \frac{9}{2 x^{\frac{1}{2}}}) + (x - 9 x^{\frac{1}{2}}) \cdot 1$

Multiply $x - 9 x^{\frac{1}{2}}$ by $1$ .

$f' (x) = x (1 - \frac{9}{2 x^{\frac{1}{2}}}) + x - 9 x^{\frac{1}{2}}$

Simplify.

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Apply the distributive property.

$f' (x) = x \cdot 1 + x (- \frac{9}{2 x^{\frac{1}{2}}}) + x - 9 x^{\frac{1}{2}}$

Combine terms.

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Multiply $x$ by $1$ .

$f' (x) = x + x (- \frac{9}{2 x^{\frac{1}{2}}}) + x - 9 x^{\frac{1}{2}}$

Combine $x$ and $\frac{9}{2 x^{\frac{1}{2}}}$ .

$f' (x) = x - \frac{x \cdot 9}{2 x^{\frac{1}{2}}} + x - 9 x^{\frac{1}{2}}$

Move $9$ to the left of $x$ .

$f' (x) = x - \frac{9 \cdot x}{2 x^{\frac{1}{2}}} + x - 9 x^{\frac{1}{2}}$

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

$f' (x) = x - \frac{9 x \cdot x^{- \frac{1}{2}}}{2} + x - 9 x^{\frac{1}{2}}$

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Move $x^{- \frac{1}{2}}$ .

$f' (x) = x - \frac{9 (x^{- \frac{1}{2}} x)}{2} + x - 9 x^{\frac{1}{2}}$

Multiply $x^{- \frac{1}{2}}$ by $x$ .

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Raise $x$ to the power of $1$ .

$f' (x) = x - \frac{9 (x^{- \frac{1}{2}} x)}{2} + x - 9 x^{\frac{1}{2}}$

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

$f' (x) = x - \frac{9 x^{- \frac{1}{2} + 1}}{2} + x - 9 x^{\frac{1}{2}}$

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

$f' (x) = x - \frac{9 x^{- \frac{1}{2} + \frac{2}{2}}}{2} + x - 9 x^{\frac{1}{2}}$

Combine the numerators over the common denominator.

$f' (x) = x - \frac{9 x^{\frac{- 1 + 2}{2}}}{2} + x - 9 x^{\frac{1}{2}}$

Add $- 1$ and $2$ .

$f' (x) = x - \frac{9 x^{\frac{1}{2}}}{2} + x - 9 x^{\frac{1}{2}}$

Add $x$ and $x$ .

$f' (x) = 2 x - \frac{9 x^{\frac{1}{2}}}{2} - 9 x^{\frac{1}{2}}$

To write $- 9 x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (x) = 2 x - \frac{9 x^{\frac{1}{2}}}{2} - 9 x^{\frac{1}{2}} \cdot \frac{2}{2}$

Combine $- 9 x^{\frac{1}{2}}$ and $\frac{2}{2}$ .

$f' (x) = 2 x - \frac{9 x^{\frac{1}{2}}}{2} + \frac{- 9 x^{\frac{1}{2}} \cdot 2}{2}$

Combine the numerators over the common denominator.

$f' (x) = 2 x + \frac{- 9 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \cdot 2}{2}$

Multiply $2$ by $- 9$ .

$f' (x) = 2 x + \frac{- 9 x^{\frac{1}{2}} - 18 x^{\frac{1}{2}}}{2}$

Subtract $18 x^{\frac{1}{2}}$ from $- 9 x^{\frac{1}{2}}$ .

$f' (x) = 2 x + \frac{- 27 x^{\frac{1}{2}}}{2}$

Move the negative in front of the fraction.

$f' (x) = 2 x - \frac{27 x^{\frac{1}{2}}}{2}$

Find the second derivative.

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By the Sum Rule, the derivative of $2 x - \frac{27 x^{\frac{1}{2}}}{2}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{27 x^{\frac{1}{2}}}{2}]$ .

$f'' (x) = \frac{d}{d x} (2 x) + \frac{d}{d x} (- \frac{27 x^{\frac{1}{2}}}{2})$

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

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

$f'' (x) = 2 \frac{d}{d x} (x) + \frac{d}{d x} (- \frac{27 x^{\frac{1}{2}}}{2})$

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

$f'' (x) = 2 \cdot 1 + \frac{d}{d x} (- \frac{27 x^{\frac{1}{2}}}{2})$

Multiply $2$ by $1$ .

$f'' (x) = 2 + \frac{d}{d x} (- \frac{27 x^{\frac{1}{2}}}{2})$

Evaluate $\frac{d}{d x} [- \frac{27 x^{\frac{1}{2}}}{2}]$ .

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Since $- \frac{27}{2}$ is constant with respect to $x$ , the derivative of $- \frac{27 x^{\frac{1}{2}}}{2}$ with respect to $x$ is $- \frac{27}{2} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$f'' (x) = 2 - \frac{27}{2} \cdot \frac{d}{d x} x^{\frac{1}{2}}$

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

$f'' (x) = 2 - \frac{27}{2} \cdot (\frac{1}{2} x^{\frac{1}{2} - 1})$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = 2 - \frac{27}{2} \cdot (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

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

$f'' (x) = 2 - \frac{27}{2} \cdot (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Combine the numerators over the common denominator.

$f'' (x) = 2 - \frac{27}{2} \cdot (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Simplify the numerator.

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Multiply $- 1$ by $2$ .

$f'' (x) = 2 - \frac{27}{2} \cdot (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Subtract $2$ from $1$ .

$f'' (x) = 2 - \frac{27}{2} \cdot (\frac{1}{2} x^{\frac{- 1}{2}})$

Move the negative in front of the fraction.

$f'' (x) = 2 - \frac{27}{2} \cdot (\frac{1}{2} x^{- \frac{1}{2}})$

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

$f'' (x) = 2 - \frac{27}{2} \cdot \frac{x^{- \frac{1}{2}}}{2}$

Multiply $\frac{x^{- \frac{1}{2}}}{2}$ by $\frac{27}{2}$ .

$f'' (x) = 2 - \frac{x^{- \frac{1}{2}} \cdot 27}{2 \cdot 2}$

Multiply $2$ by $2$ .

$f'' (x) = 2 - \frac{x^{- \frac{1}{2}} \cdot 27}{4}$

Move $27$ to the left of $x^{- \frac{1}{2}}$ .

$f'' (x) = 2 - \frac{27 \cdot x^{- \frac{1}{2}}}{4}$

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

$f'' (x) = 2 - \frac{27}{4 x^{\frac{1}{2}}}$

The second derivative of $f (x)$ with respect to $x$ is $2 - \frac{27}{4 x^{\frac{1}{2}}}$ .

$2 - \frac{27}{4 x^{\frac{1}{2}}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $2 - \frac{27}{4 x^{\frac{1}{2}}} = 0$ .

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Set the second derivative equal to $0$ .

$2 - \frac{27}{4 x^{\frac{1}{2}}} = 0$

Subtract $2$ from both sides of the equation.

$- \frac{27}{4 x^{\frac{1}{2}}} = - 2$

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$4 x^{\frac{1}{2}}, 1$

The LCM of one and any expression is the expression.

$4 x^{\frac{1}{2}}$

Multiply each term in $- \frac{27}{4 x^{\frac{1}{2}}} = - 2$ by $4 x^{\frac{1}{2}}$ to eliminate the fractions.

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Multiply each term in $- \frac{27}{4 x^{\frac{1}{2}}} = - 2$ by $4 x^{\frac{1}{2}}$ .

$- \frac{27}{4 x^{\frac{1}{2}}} (4 x^{\frac{1}{2}}) = - 2 (4 x^{\frac{1}{2}})$

Simplify the left side.

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Cancel the common factor of $4 x^{\frac{1}{2}}$ .

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Move the leading negative in $- \frac{27}{4 x^{\frac{1}{2}}}$ into the numerator.

$\frac{- 27}{4 x^{\frac{1}{2}}} (4 x^{\frac{1}{2}}) = - 2 (4 x^{\frac{1}{2}})$

Cancel the common factor.

$\frac{- 27}{4 x^{\frac{1}{2}}} (4 x^{\frac{1}{2}}) = - 2 (4 x^{\frac{1}{2}})$

Rewrite the expression.

$- 27 = - 2 (4 x^{\frac{1}{2}})$

Simplify the right side.

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Multiply $4$ by $- 2$ .

$- 27 = - 8 x^{\frac{1}{2}}$

Solve the equation.

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Rewrite the equation as $- 8 x^{\frac{1}{2}} = - 27$ .

$- 8 x^{\frac{1}{2}} = - 27$

Divide each term in $- 8 x^{\frac{1}{2}} = - 27$ by $- 8$ and simplify.

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Divide each term in $- 8 x^{\frac{1}{2}} = - 27$ by $- 8$ .

$\frac{- 8 x^{\frac{1}{2}}}{- 8} = \frac{- 27}{- 8}$

Simplify the left side.

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Cancel the common factor.

$\frac{- 8 x^{\frac{1}{2}}}{- 8} = \frac{- 27}{- 8}$

Divide $x^{\frac{1}{2}}$ by $1$ .

$x^{\frac{1}{2}} = \frac{- 27}{- 8}$

Simplify the right side.

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Dividing two negative values results in a positive value.

$x^{\frac{1}{2}} = \frac{27}{8}$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = {(\frac{27}{8})}^{2}$

Simplify the exponent.

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Simplify the left side.

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Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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

$x^{\frac{1}{2} \cdot 2} = {(\frac{27}{8})}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\frac{27}{8})}^{2}$

Rewrite the expression.

$x^{1} = {(\frac{27}{8})}^{2}$

Simplify.

$x = {(\frac{27}{8})}^{2}$

Simplify the right side.

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Simplify ${(\frac{27}{8})}^{2}$ .

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Apply the product rule to $\frac{27}{8}$ .

$x = \frac{27^{2}}{8^{2}}$

Raise $27$ to the power of $2$ .

$x = \frac{729}{8^{2}}$

Raise $8$ to the power of $2$ .

$x = \frac{729}{64}$

Step 3

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

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Substitute $\frac{729}{64}$ in $f (x) = x (x - 9 \sqrt{x})$ to find the value of $y$ .

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Replace the variable $x$ with $\frac{729}{64}$ in the expression.

$f (\frac{729}{64}) = (\frac{729}{64}) ((\frac{729}{64}) - 9 \sqrt{\frac{729}{64}})$

Simplify the result.

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Simplify each term.

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Rewrite $\sqrt{\frac{729}{64}}$ as $\frac{\sqrt{729}}{\sqrt{64}}$ .

$f (\frac{729}{64}) = \frac{729}{64} \cdot (\frac{729}{64} - 9 \frac{\sqrt{729}}{\sqrt{64}})$

Simplify the numerator.

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Rewrite $729$ as $27^{2}$ .

$f (\frac{729}{64}) = \frac{729}{64} \cdot (\frac{729}{64} - 9 \frac{\sqrt{27^{2}}}{\sqrt{64}})$

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

$f (\frac{729}{64}) = \frac{729}{64} \cdot (\frac{729}{64} - 9 \frac{27}{\sqrt{64}})$

Simplify the denominator.

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Rewrite $64$ as $8^{2}$ .

$f (\frac{729}{64}) = \frac{729}{64} \cdot (\frac{729}{64} - 9 \frac{27}{\sqrt{8^{2}}})$

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

$f (\frac{729}{64}) = \frac{729}{64} \cdot (\frac{729}{64} - 9 (\frac{27}{8}))$

Multiply $- 9 (\frac{27}{8})$ .

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Combine $- 9$ and $\frac{27}{8}$ .

$f (\frac{729}{64}) = \frac{729}{64} \cdot (\frac{729}{64} + \frac{- 9 \cdot 27}{8})$

Multiply $- 9$ by $27$ .

$f (\frac{729}{64}) = \frac{729}{64} \cdot (\frac{729}{64} + \frac{- 243}{8})$

Move the negative in front of the fraction.

$f (\frac{729}{64}) = \frac{729}{64} \cdot (\frac{729}{64} - \frac{243}{8})$

To write $- \frac{243}{8}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (\frac{729}{64}) = \frac{729}{64} \cdot (\frac{729}{64} - \frac{243}{8} \cdot \frac{8}{8})$

Write each expression with a common denominator of $64$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{243}{8}$ by $\frac{8}{8}$ .

$f (\frac{729}{64}) = \frac{729}{64} \cdot (\frac{729}{64} - \frac{243 \cdot 8}{8 \cdot 8})$

Multiply $8$ by $8$ .

$f (\frac{729}{64}) = \frac{729}{64} \cdot (\frac{729}{64} - \frac{243 \cdot 8}{64})$

Combine the numerators over the common denominator.

$f (\frac{729}{64}) = \frac{729}{64} \cdot \frac{729 - 243 \cdot 8}{64}$

Simplify the numerator.

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Multiply $- 243$ by $8$ .

$f (\frac{729}{64}) = \frac{729}{64} \cdot \frac{729 - 1944}{64}$

Subtract $1944$ from $729$ .

$f (\frac{729}{64}) = \frac{729}{64} \cdot \frac{- 1215}{64}$

Move the negative in front of the fraction.

$f (\frac{729}{64}) = \frac{729}{64} \cdot (- \frac{1215}{64})$

Multiply $\frac{729}{64} (- \frac{1215}{64})$ .

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Multiply $\frac{729}{64}$ by $\frac{1215}{64}$ .

$f (\frac{729}{64}) = - \frac{729 \cdot 1215}{64 \cdot 64}$

Multiply $729$ by $1215$ .

$f (\frac{729}{64}) = - \frac{885735}{64 \cdot 64}$

Multiply $64$ by $64$ .

$f (\frac{729}{64}) = - \frac{885735}{4096}$

The final answer is $- \frac{885735}{4096}$ .

$- \frac{885735}{4096}$

The point found by substituting $\frac{729}{64}$ in $f (x) = x (x - 9 \sqrt{x})$ is $(\frac{729}{64}, - \frac{885735}{4096})$ . This point can be an inflection point.

$(\frac{729}{64}, - \frac{885735}{4096})$

Step 4

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

$(- \infty, \frac{729}{64}) \cup (\frac{729}{64}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{729}{64})$ into the second derivative to determine if it is increasing or decreasing.

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Replace the variable $x$ with $11.290625$ in the expression.

$f'' (11.290625) = 2 - \frac{27}{4 {(11.290625)}^{\frac{1}{2}}}$

Simplify the result.

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Simplify each term.

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Simplify the denominator.

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Rewrite $11.290625$ as ${3.36015252}^{2}$ .

$f'' (11.290625) = 2 - \frac{27}{4 \cdot {({3.36015252}^{2})}^{\frac{1}{2}}}$

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

$f'' (11.290625) = 2 - \frac{27}{4 \cdot {3.36015252}^{2 (\frac{1}{2})}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$f'' (11.290625) = 2 - \frac{27}{4 \cdot {3.36015252}^{2 (\frac{1}{2})}}$

Rewrite the expression.

$f'' (11.290625) = 2 - \frac{27}{4 \cdot 3.36015252}$

Evaluate the exponent.

$f'' (11.290625) = 2 - \frac{27}{4 \cdot 3.36015252}$

Multiply $4$ by $3.36015252$ .

$f'' (11.290625) = 2 - \frac{27}{13.4406101}$

Divide $27$ by $13.4406101$ .

$f'' (11.290625) = 2 - 1 \cdot 2.00883738$

Multiply $- 1$ by $2.00883738$ .

$f'' (11.290625) = 2 - 2.00883738$

Subtract $2.00883738$ from $2$ .

$f'' (11.290625) = - 0.00883738$

The final answer is $- 0.00883738$ .

$- 0.00883738$

At $11.290625$ , the second derivative is $- 0.00883738$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, \frac{729}{64})$

Decreasing on $(- \infty, \frac{729}{64})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(\frac{729}{64}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Replace the variable $x$ with $11.490625$ in the expression.

$f'' (11.490625) = 2 - \frac{27}{4 {(11.490625)}^{\frac{1}{2}}}$

Simplify the result.

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Simplify each term.

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Simplify the denominator.

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Rewrite $11.490625$ as ${3.38978244}^{2}$ .

$f'' (11.490625) = 2 - \frac{27}{4 \cdot {({3.38978244}^{2})}^{\frac{1}{2}}}$

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

$f'' (11.490625) = 2 - \frac{27}{4 \cdot {3.38978244}^{2 (\frac{1}{2})}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$f'' (11.490625) = 2 - \frac{27}{4 \cdot {3.38978244}^{2 (\frac{1}{2})}}$

Rewrite the expression.

$f'' (11.490625) = 2 - \frac{27}{4 \cdot 3.38978244}$

Evaluate the exponent.

$f'' (11.490625) = 2 - \frac{27}{4 \cdot 3.38978244}$

Multiply $4$ by $3.38978244$ .

$f'' (11.490625) = 2 - \frac{27}{13.55912976}$

Divide $27$ by $13.55912976$ .

$f'' (11.490625) = 2 - 1 \cdot 1.99127823$

Multiply $- 1$ by $1.99127823$ .

$f'' (11.490625) = 2 - 1.99127823$

Subtract $1.99127823$ from $2$ .

$f'' (11.490625) = 0.00872176$

The final answer is $0.00872176$ .

$0.00872176$

At $11.490625$ , the second derivative is $0.00872176$ . Since this is positive, the second derivative is increasing on the interval $(\frac{729}{64}, \infty)$ .

Increasing on $(\frac{729}{64}, \infty)$ 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 $(\frac{729}{64}, - \frac{885735}{4096})$ .

$(\frac{729}{64}, - \frac{885735}{4096})$

Step 8