Find the Inflection Points f(x)=x-6 square root of x

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

Step 1

Find the second derivative.

Find the first derivative.

Differentiate.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

Factor $2$ out of $- 6$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Find the second derivative.

Differentiate.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $- 1$ .

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

Move the negative in front of the fraction.

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

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

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

Multiply $- 1$ by $- 3$ .

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

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

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

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

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

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

Set the second derivative equal to $0$ .

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

Set the numerator equal to zero.

$3 = 0$

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

Find the Inflection Points f(x)=x-6 square root of x

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