Calculus Examples

$f (x) = x^{2} | x |$

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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 $f (x) = x^{2}$ and $g (x) = | x |$ .

$x^{2} \frac{d}{d x} [| x |] + | x | \frac{d}{d x} [x^{2}]$

Step 1.1.1.2

The derivative of $| x |$ with respect to $x$ is $\frac{x}{| x |}$ .

$x^{2} \frac{x}{| x |} + | x | \frac{d}{d x} [x^{2}]$

Step 1.1.1.3

Combine $x^{2}$ and $\frac{x}{| x |}$ .

$\frac{x^{2} x}{| x |} + | x | \frac{d}{d x} [x^{2}]$

Step 1.1.1.4

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.1.4.1

Multiply $x^{2}$ by $x$ .

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Step 1.1.1.4.1.1

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

$\frac{x^{2} x^{1}}{| x |} + | x | \frac{d}{d x} [x^{2}]$

Step 1.1.1.4.1.2

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

$\frac{x^{2 + 1}}{| x |} + | x | \frac{d}{d x} [x^{2}]$

Step 1.1.1.4.2

Add $2$ and $1$ .

$\frac{x^{3}}{| x |} + | x | \frac{d}{d x} [x^{2}]$

Step 1.1.1.5

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

$\frac{x^{3}}{| x |} + | x | (2 x)$

Step 1.1.1.6

Reorder terms.

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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

$\frac{d}{d x} [\frac{x^{3}}{| x |}] + \frac{d}{d x} [2 x | x |]$

Step 1.1.2.2

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

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Step 1.1.2.2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{3}$ and $g (x) = | x |$ .

$\frac{| x | \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [| x |]}{{| x |}^{2}} + \frac{d}{d x} [2 x | x |]$

Step 1.1.2.2.2

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

$\frac{| x | (3 x^{2}) - x^{3} \frac{d}{d x} [| x |]}{{| x |}^{2}} + \frac{d}{d x} [2 x | x |]$

Step 1.1.2.2.3

The derivative of $| x |$ with respect to $x$ is $\frac{x}{| x |}$ .

$\frac{| x | (3 x^{2}) - x^{3} \frac{x}{| x |}}{{| x |}^{2}} + \frac{d}{d x} [2 x | x |]$

Step 1.1.2.2.4

Combine $\frac{x}{| x |}$ and $x^{3}$ .

$\frac{| x | (3 x^{2}) - \frac{x \cdot x^{3}}{| x |}}{{| x |}^{2}} + \frac{d}{d x} [2 x | x |]$

Step 1.1.2.2.5

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 1.1.2.2.5.1

Multiply $x$ by $x^{3}$ .

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Step 1.1.2.2.5.1.1

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

$\frac{| x | (3 x^{2}) - \frac{x^{1} x^{3}}{| x |}}{{| x |}^{2}} + \frac{d}{d x} [2 x | x |]$

Step 1.1.2.2.5.1.2

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

$\frac{| x | (3 x^{2}) - \frac{x^{1 + 3}}{| x |}}{{| x |}^{2}} + \frac{d}{d x} [2 x | x |]$

Step 1.1.2.2.5.2

Add $1$ and $3$ .

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + \frac{d}{d x} [2 x | x |]$

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + 2 \frac{d}{d x} [x | x |]$

Step 1.1.2.3.2

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 $g (x) = | x |$ .

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + 2 (x \frac{d}{d x} [| x |] + | x | \frac{d}{d x} [x])$

Step 1.1.2.3.3

The derivative of $| x |$ with respect to $x$ is $\frac{x}{| x |}$ .

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + 2 (x \frac{x}{| x |} + | x | \frac{d}{d x} [x])$

Step 1.1.2.3.4

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

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + 2 (x \frac{x}{| x |} + | x | \cdot 1)$

Step 1.1.2.3.5

Combine $x$ and $\frac{x}{| x |}$ .

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + 2 (\frac{x \cdot x}{| x |} + | x | \cdot 1)$

Step 1.1.2.3.6

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

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + 2 (\frac{x^{1} x}{| x |} + | x | \cdot 1)$

Step 1.1.2.3.7

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

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + 2 (\frac{x^{1} x^{1}}{| x |} + | x | \cdot 1)$

Step 1.1.2.3.8

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

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + 2 (\frac{x^{1 + 1}}{| x |} + | x | \cdot 1)$

Step 1.1.2.3.9

Add $1$ and $1$ .

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + 2 (\frac{x^{2}}{| x |} + | x | \cdot 1)$

Step 1.1.2.3.10

Multiply $| x |$ by $1$ .

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + 2 (\frac{x^{2}}{| x |} + | x |)$

Step 1.1.2.4

Simplify.

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Step 1.1.2.4.1

Apply the distributive property.

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + 2 \frac{x^{2}}{| x |} + 2 | x |$

Step 1.1.2.4.2

Combine terms.

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Step 1.1.2.4.2.1

Combine $2$ and $\frac{x^{2}}{| x |}$ .

$\frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + \frac{2 x^{2}}{| x |} + 2 | x |$

Step 1.1.2.4.2.2

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

$\frac{2 x^{2}}{| x |} + \frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |}}{{| x |}^{2}} + \frac{2 | x | {| x |}^{2}}{{| x |}^{2}}$

Step 1.1.2.4.2.3

Combine the numerators over the common denominator.

$\frac{2 x^{2}}{| x |} + \frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |} + 2 | x | {| x |}^{2}}{{| x |}^{2}}$

Step 1.1.2.4.2.4

Multiply $| x |$ by ${| x |}^{2}$ by adding the exponents.

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Step 1.1.2.4.2.4.1

Move ${| x |}^{2}$ .

$\frac{2 x^{2}}{| x |} + \frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |} + 2 ({| x |}^{2} | x |)}{{| x |}^{2}}$

Step 1.1.2.4.2.4.2

Multiply ${| x |}^{2}$ by $| x |$ .

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Step 1.1.2.4.2.4.2.1

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

$\frac{2 x^{2}}{| x |} + \frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |} + 2 ({| x |}^{2} {| x |}^{1})}{{| x |}^{2}}$

Step 1.1.2.4.2.4.2.2

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

$\frac{2 x^{2}}{| x |} + \frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |} + 2 {| x |}^{2 + 1}}{{| x |}^{2}}$

Step 1.1.2.4.2.4.3

Add $2$ and $1$ .

$\frac{2 x^{2}}{| x |} + \frac{| x | (3 x^{2}) - \frac{x^{4}}{| x |} + 2 {| x |}^{3}}{{| x |}^{2}}$

Step 1.1.2.4.3

Simplify each term.

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Step 1.1.2.4.3.1

Simplify the numerator.

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Step 1.1.2.4.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2}}{| x |} + \frac{3 | x | x^{2} - \frac{x^{4}}{| x |} + 2 {| x |}^{3}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.2

Reorder terms.

$\frac{2 x^{2}}{| x |} + \frac{- \frac{x^{4}}{| x |} + 2 {| x |}^{3} + 3 x^{2} | x |}{{| x |}^{2}}$

Step 1.1.2.4.3.1.3

To write $2 {| x |}^{3}$ as a fraction with a common denominator, multiply by $\frac{| x |}{| x |}$ .

$\frac{2 x^{2}}{| x |} + \frac{- \frac{x^{4}}{| x |} + \frac{2 {| x |}^{3} | x |}{| x |} + 3 x^{2} | x |}{{| x |}^{2}}$

Step 1.1.2.4.3.1.4

Combine the numerators over the common denominator.

$\frac{2 x^{2}}{| x |} + \frac{\frac{- x^{4} + 2 {| x |}^{3} | x |}{| x |} + 3 x^{2} | x |}{{| x |}^{2}}$

Step 1.1.2.4.3.1.5

Simplify the numerator.

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Step 1.1.2.4.3.1.5.1

Multiply ${| x |}^{3}$ by $| x |$ by adding the exponents.

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Step 1.1.2.4.3.1.5.1.1

Move $| x |$ .

$\frac{2 x^{2}}{| x |} + \frac{\frac{- x^{4} + 2 (| x | {| x |}^{3})}{| x |} + 3 x^{2} | x |}{{| x |}^{2}}$

Step 1.1.2.4.3.1.5.1.2

Multiply $| x |$ by ${| x |}^{3}$ .

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Step 1.1.2.4.3.1.5.1.2.1

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

$\frac{2 x^{2}}{| x |} + \frac{\frac{- x^{4} + 2 ({| x |}^{1} {| x |}^{3})}{| x |} + 3 x^{2} | x |}{{| x |}^{2}}$

Step 1.1.2.4.3.1.5.1.2.2

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

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

Step 1.1.2.4.3.1.5.1.3

Add $1$ and $3$ .

$\frac{2 x^{2}}{| x |} + \frac{\frac{- x^{4} + 2 {| x |}^{4}}{| x |} + 3 x^{2} | x |}{{| x |}^{2}}$

Step 1.1.2.4.3.1.5.2

Remove the absolute value in ${| x |}^{4}$ because exponentiations with even powers are always positive.

$\frac{2 x^{2}}{| x |} + \frac{\frac{- x^{4} + 2 x^{4}}{| x |} + 3 x^{2} | x |}{{| x |}^{2}}$

Step 1.1.2.4.3.1.5.3

Add $- x^{4}$ and $2 x^{4}$ .

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{4}}{| x |} + 3 x^{2} | x |}{{| x |}^{2}}$

Step 1.1.2.4.3.1.6

To write $3 x^{2} | x |$ as a fraction with a common denominator, multiply by $\frac{| x |}{| x |}$ .

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{4}}{| x |} + \frac{3 x^{2} | x | | x |}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.7

Combine the numerators over the common denominator.

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{4} + 3 x^{2} | x | | x |}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.8

Simplify the numerator.

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Step 1.1.2.4.3.1.8.1

Factor $x^{2}$ out of $x^{4} + 3 x^{2} | x | | x |$ .

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Step 1.1.2.4.3.1.8.1.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 1.1.2.4.3.1.8.1.2

Factor $x^{2}$ out of $3 x^{2} | x | | x |$ .

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{2} x^{2} + x^{2} (3 | x | | x |)}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.8.1.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (3 | x | | x |)$ .

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{2} (x^{2} + 3 | x | | x |)}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.8.2

Multiply $3 | x | | x |$ .

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Step 1.1.2.4.3.1.8.2.1

To multiply absolute values, multiply the terms inside each absolute value.

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{2} (x^{2} + 3 | x \cdot x |)}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.8.2.2

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

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{2} (x^{2} + 3 | x^{1} x |)}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.8.2.3

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

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{2} (x^{2} + 3 | x^{1} x^{1} |)}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.8.2.4

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

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{2} (x^{2} + 3 | x^{1 + 1} |)}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.8.2.5

Add $1$ and $1$ .

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{2} (x^{2} + 3 | x^{2} |)}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.8.3

Remove non-negative terms from the absolute value.

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{2} (x^{2} + 3 x^{2})}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.8.4

Add $x^{2}$ and $3 x^{2}$ .

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{2} \cdot 4 x^{2}}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.8.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.1.2.4.3.1.8.5.1

Move $x^{2}$ .

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{2} x^{2} \cdot 4}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.8.5.2

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

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{2 + 2} \cdot 4}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.8.5.3

Add $2$ and $2$ .

$\frac{2 x^{2}}{| x |} + \frac{\frac{x^{4} \cdot 4}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.1.9

Move $4$ to the left of $x^{4}$ .

$\frac{2 x^{2}}{| x |} + \frac{\frac{4 x^{4}}{| x |}}{{| x |}^{2}}$

Step 1.1.2.4.3.2

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

$\frac{2 x^{2}}{| x |} + \frac{\frac{4 x^{4}}{| x |}}{x^{2}}$

Step 1.1.2.4.3.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x^{2}}{| x |} + \frac{4 x^{4}}{| x |} \cdot \frac{1}{x^{2}}$

Step 1.1.2.4.3.4

Combine.

$\frac{2 x^{2}}{| x |} + \frac{4 x^{4} \cdot 1}{| x | x^{2}}$

Step 1.1.2.4.3.5

Cancel the common factor of $x^{4}$ and $x^{2}$ .

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Step 1.1.2.4.3.5.1

Factor $x^{2}$ out of $4 x^{4} \cdot 1$ .

$\frac{2 x^{2}}{| x |} + \frac{x^{2} (4 x^{2} \cdot 1)}{| x | x^{2}}$

Step 1.1.2.4.3.5.2

Cancel the common factors.

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Step 1.1.2.4.3.5.2.1

Factor $x^{2}$ out of $| x | x^{2}$ .

$\frac{2 x^{2}}{| x |} + \frac{x^{2} (4 x^{2} \cdot 1)}{x^{2} | x |}$

Step 1.1.2.4.3.5.2.2

Cancel the common factor.

$\frac{2 x^{2}}{| x |} + \frac{x^{2} (4 x^{2} \cdot 1)}{x^{2} | x |}$

Step 1.1.2.4.3.5.2.3

Rewrite the expression.

$\frac{2 x^{2}}{| x |} + \frac{4 x^{2} \cdot 1}{| x |}$

Step 1.1.2.4.3.6

Multiply $4$ by $1$ .

$\frac{2 x^{2}}{| x |} + \frac{4 x^{2}}{| x |}$

Step 1.1.2.4.4

Combine the numerators over the common denominator.

$\frac{2 x^{2} + 4 x^{2}}{| x |}$

Step 1.1.2.4.5

Add $2 x^{2}$ and $4 x^{2}$ .

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

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{6 x^{2}}{| x |}$ .

$\frac{6 x^{2}}{| x |}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $\frac{6 x^{2}}{| x |} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$\frac{6 x^{2}}{| x |} = 0$

Step 1.2.2

Set the numerator equal to zero.

$6 x^{2} = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

Divide each term in $6 x^{2} = 0$ by $6$ and simplify.

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Step 1.2.3.1.1

Divide each term in $6 x^{2} = 0$ by $6$ .

$\frac{6 x^{2}}{6} = \frac{0}{6}$

Step 1.2.3.1.2

Simplify the left side.

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Step 1.2.3.1.2.1

Cancel the common factor of $6$ .

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Step 1.2.3.1.2.1.1

Cancel the common factor.

$\frac{6 x^{2}}{6} = \frac{0}{6}$

Step 1.2.3.1.2.1.2

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

$x^{2} = \frac{0}{6}$

Step 1.2.3.1.3

Simplify the right side.

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Step 1.2.3.1.3.1

Divide $0$ by $6$ .

$x^{2} = 0$

Step 1.2.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 1.2.3.3

Simplify $\pm \sqrt{0}$ .

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Step 1.2.3.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.3.3.2

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

$x = \pm 0$

Step 1.2.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.4

Exclude the solutions that do not make $\frac{6 x^{2}}{| x |} = 0$ true.

$No solution$

Step 2

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

The graph is concave up because the second derivative is positive.

The graph is concave up

Step 4