Algebra Examples

$f (x) = x - 6 \sqrt{x}$

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

Differentiate.

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Step 1.1.1.1

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

$f' (x) = \frac{d}{d x} (x) + \frac{d}{d x} (- 6 \sqrt{x})$

Step 1.1.1.2

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

$f' (x) = 1 + \frac{d}{d x} (- 6 \sqrt{x})$

Step 1.1.2

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

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Step 1.1.2.1

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

$f' (x) = 1 + \frac{d}{d x} (- 6 x^{\frac{1}{2}})$

Step 1.1.2.2

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{\frac{1}{2}}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$f' (x) = 1 - 6 \frac{d}{d x} x^{\frac{1}{2}}$

Step 1.1.2.3

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

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

Step 1.1.2.4

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

$f' (x) = 1 - 6 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 1.1.2.5

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

$f' (x) = 1 - 6 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 1.1.2.6

Combine the numerators over the common denominator.

$f' (x) = 1 - 6 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 1.1.2.7

Simplify the numerator.

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Step 1.1.2.7.1

Multiply $- 1$ by $2$ .

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

Step 1.1.2.7.2

Subtract $2$ from $1$ .

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

Step 1.1.2.8

Move the negative in front of the fraction.

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

Step 1.1.2.9

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

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

Step 1.1.2.10

Combine $- 6$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$f' (x) = 1 + \frac{- 6 x^{- \frac{1}{2}}}{2}$

Step 1.1.2.11

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

$f' (x) = 1 + \frac{- 6}{2 x^{\frac{1}{2}}}$

Step 1.1.2.12

Factor $2$ out of $- 6$ .

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

Step 1.1.2.13

Cancel the common factors.

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Step 1.1.2.13.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 1.1.2.13.2

Cancel the common factor.

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

Step 1.1.2.13.3

Rewrite the expression.

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

Step 1.1.2.14

Move the negative in front of the fraction.

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

Step 1.2

Find the second derivative.

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Step 1.2.1

Differentiate.

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Step 1.2.1.1

By the Sum Rule, the derivative of $1 - \frac{3}{x^{\frac{1}{2}}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{3}{x^{\frac{1}{2}}}]$ .

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

Step 1.2.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = 0 + \frac{d}{d x} (- \frac{3}{x^{\frac{1}{2}}})$

Step 1.2.2

Evaluate $\frac{d}{d x} [- \frac{3}{x^{\frac{1}{2}}}]$ .

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Step 1.2.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3}{x^{\frac{1}{2}}}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

$f'' (x) = 0 - 3 \frac{d}{d x} \frac{1}{x^{\frac{1}{2}}}$

Step 1.2.2.2

Rewrite $\frac{1}{x^{\frac{1}{2}}}$ as ${(x^{\frac{1}{2}})}^{- 1}$ .

$f'' (x) = 0 - 3 \frac{d}{d x} {(x^{\frac{1}{2}})}^{- 1}$

Step 1.2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{1}{2}}$ .

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Step 1.2.2.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}}$ .

$f'' (x) = 0 - 3 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{\frac{1}{2}}))$

Step 1.2.2.3.2

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

$f'' (x) = 0 - 3 (- u^{- 2} \frac{d}{d x} x^{\frac{1}{2}})$

Step 1.2.2.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

$f'' (x) = 0 - 3 (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} x^{\frac{1}{2}})$

Step 1.2.2.4

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

$f'' (x) = 0 - 3 (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 1.2.2.5

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

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Step 1.2.2.5.1

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

$f'' (x) = 0 - 3 (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 1.2.2.5.2

Cancel the common factor of $2$ .

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Step 1.2.2.5.2.1

Factor $2$ out of $- 2$ .

$f'' (x) = 0 - 3 (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 1.2.2.5.2.2

Cancel the common factor.

$f'' (x) = 0 - 3 (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 1.2.2.5.2.3

Rewrite the expression.

$f'' (x) = 0 - 3 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 1.2.2.6

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

$f'' (x) = 0 - 3 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 1.2.2.7

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

$f'' (x) = 0 - 3 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 1.2.2.8

Combine the numerators over the common denominator.

$f'' (x) = 0 - 3 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 1.2.2.9

Simplify the numerator.

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Step 1.2.2.9.1

Multiply $- 1$ by $2$ .

$f'' (x) = 0 - 3 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 1.2.2.9.2

Subtract $2$ from $1$ .

$f'' (x) = 0 - 3 (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 1.2.2.10

Move the negative in front of the fraction.

$f'' (x) = 0 - 3 (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 1.2.2.11

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

$f'' (x) = 0 - 3 (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2})$

Step 1.2.2.12

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

$f'' (x) = 0 - 3 (- \frac{x^{- \frac{1}{2}} x^{- 1}}{2})$

Step 1.2.2.13

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

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Step 1.2.2.13.1

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

$f'' (x) = 0 - 3 (- \frac{x^{- \frac{1}{2} - 1}}{2})$

Step 1.2.2.13.2

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

$f'' (x) = 0 - 3 (- \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2})$

Step 1.2.2.13.3

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

$f'' (x) = 0 - 3 (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2})$

Step 1.2.2.13.4

Combine the numerators over the common denominator.

$f'' (x) = 0 - 3 (- \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2})$

Step 1.2.2.13.5

Simplify the numerator.

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Step 1.2.2.13.5.1

Multiply $- 1$ by $2$ .

$f'' (x) = 0 - 3 (- \frac{x^{\frac{- 1 - 2}{2}}}{2})$

Step 1.2.2.13.5.2

Subtract $2$ from $- 1$ .

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

Step 1.2.2.13.6

Move the negative in front of the fraction.

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

Step 1.2.2.14

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

$f'' (x) = 0 - 3 (- \frac{1}{2 x^{\frac{3}{2}}})$

Step 1.2.2.15

Multiply $- 1$ by $- 3$ .

$f'' (x) = 0 + 3 (\frac{1}{2 x^{\frac{3}{2}}})$

Step 1.2.2.16

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

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

Step 1.2.3

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

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

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{3}{2 x^{\frac{3}{2}}}$ .

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

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{3}{2 x^{\frac{3}{2}}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$\frac{3}{2 x^{\frac{3}{2}}} = 0$

Step 2.2

Set the numerator equal to zero.

$3 = 0$

Step 2.3

Since $3 \neq 0$ , there are no solutions.

No solution

Step 3

No values found that can make the second derivative equal to $0$ .

No Inflection Points