Calculus Examples

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

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 x^{\frac{2}{3}} - 2 x = 0$ .

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

Step 1.2.2

Factor $x^{\frac{2}{3}}$ out of $3 x^{\frac{2}{3}} - 2 x$ .

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Step 1.2.2.1

Factor $x^{\frac{2}{3}}$ out of $3 x^{\frac{2}{3}}$ .

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

Step 1.2.2.2

Factor $x^{\frac{2}{3}}$ out of $- 2 x$ .

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

Step 1.2.2.3

Factor $x^{\frac{2}{3}}$ out of $x^{\frac{2}{3}} \cdot 3 + x^{\frac{2}{3}} (- 2 x^{\frac{1}{3}})$ .

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

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

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

$3 - 2 x^{\frac{1}{3}} = 0$

Step 1.2.4

Set $x^{\frac{2}{3}}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x^{\frac{2}{3}}$ equal to $0$ .

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

Step 1.2.4.2

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

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Step 1.2.4.2.1

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 0^{\frac{3}{2}}$

Step 1.2.4.2.2

Simplify the exponent.

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Step 1.2.4.2.2.1

Simplify the left side.

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Step 1.2.4.2.2.1.1

Simplify ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 1.2.4.2.2.1.1.1

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

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Step 1.2.4.2.2.1.1.1.1

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

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

Step 1.2.4.2.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.1.1.1.2.1

Cancel the common factor.

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

Step 1.2.4.2.2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm 0^{\frac{3}{2}}$

Step 1.2.4.2.2.1.1.1.3

Cancel the common factor of $3$ .

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Step 1.2.4.2.2.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm 0^{\frac{3}{2}}$

Step 1.2.4.2.2.1.1.1.3.2

Rewrite the expression.

$x^{1} = \pm 0^{\frac{3}{2}}$

Step 1.2.4.2.2.1.1.2

Simplify.

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

Step 1.2.4.2.2.2

Simplify the right side.

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Step 1.2.4.2.2.2.1

Simplify $\pm 0^{\frac{3}{2}}$ .

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Step 1.2.4.2.2.2.1.1

Simplify the expression.

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Step 1.2.4.2.2.2.1.1.1

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

$x = \pm {(0^{2})}^{\frac{3}{2}}$

Step 1.2.4.2.2.2.1.1.2

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

$x = \pm 0^{2 (\frac{3}{2})}$

Step 1.2.4.2.2.2.1.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.2.1.2.1

Cancel the common factor.

$x = \pm 0^{2 (\frac{3}{2})}$

Step 1.2.4.2.2.2.1.2.2

Rewrite the expression.

$x = \pm 0^{3}$

Step 1.2.4.2.2.2.1.3

Raising $0$ to any positive power yields $0$ .

$x = \pm 0$

Step 1.2.4.2.2.2.1.4

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.5

Set $3 - 2 x^{\frac{1}{3}}$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $3 - 2 x^{\frac{1}{3}}$ equal to $0$ .

$3 - 2 x^{\frac{1}{3}} = 0$

Step 1.2.5.2

Solve $3 - 2 x^{\frac{1}{3}} = 0$ for $x$ .

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Step 1.2.5.2.1

Subtract $3$ from both sides of the equation.

$- 2 x^{\frac{1}{3}} = - 3$

Step 1.2.5.2.2

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

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

Step 1.2.5.2.3

Simplify the exponent.

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Step 1.2.5.2.3.1

Simplify the left side.

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Step 1.2.5.2.3.1.1

Simplify ${(- 2 x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.5.2.3.1.1.1

Apply the product rule to $- 2 x^{\frac{1}{3}}$ .

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

Step 1.2.5.2.3.1.1.2

Raise $- 2$ to the power of $3$ .

$- 8 {(x^{\frac{1}{3}})}^{3} = {(- 3)}^{3}$

Step 1.2.5.2.3.1.1.3

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

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Step 1.2.5.2.3.1.1.3.1

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

$- 8 x^{\frac{1}{3} \cdot 3} = {(- 3)}^{3}$

Step 1.2.5.2.3.1.1.3.2

Cancel the common factor of $3$ .

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Step 1.2.5.2.3.1.1.3.2.1

Cancel the common factor.

$- 8 x^{\frac{1}{3} \cdot 3} = {(- 3)}^{3}$

Step 1.2.5.2.3.1.1.3.2.2

Rewrite the expression.

$- 8 x^{1} = {(- 3)}^{3}$

Step 1.2.5.2.3.1.1.4

Simplify.

$- 8 x = {(- 3)}^{3}$

Step 1.2.5.2.3.2

Simplify the right side.

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Step 1.2.5.2.3.2.1

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

$- 8 x = - 27$

Step 1.2.5.2.4

Divide each term in $- 8 x = - 27$ by $- 8$ and simplify.

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Step 1.2.5.2.4.1

Divide each term in $- 8 x = - 27$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 27}{- 8}$

Step 1.2.5.2.4.2

Simplify the left side.

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Step 1.2.5.2.4.2.1

Cancel the common factor of $- 8$ .

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Step 1.2.5.2.4.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- 27}{- 8}$

Step 1.2.5.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 27}{- 8}$

Step 1.2.5.2.4.3

Simplify the right side.

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Step 1.2.5.2.4.3.1

Dividing two negative values results in a positive value.

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

Step 1.2.6

The final solution is all the values that make $x^{\frac{2}{3}} (3 - 2 x^{\frac{1}{3}}) = 0$ true.

$x = 0, \frac{27}{8}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\frac{27}{8}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 3 {(0)}^{\frac{2}{3}} - 2 \cdot 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 3 \cdot 0^{\frac{2}{3}} - 2 \cdot 0$

Step 2.2.2

Remove parentheses.

$y = 3 {(0)}^{\frac{2}{3}} - 2 \cdot 0$

Step 2.2.3

Simplify $3 {(0)}^{\frac{2}{3}} - 2 \cdot 0$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Rewrite $0$ as $0^{3}$ .

$y = 3 {(0^{3})}^{\frac{2}{3}} - 2 \cdot 0$

Step 2.2.3.1.2

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

$y = 3 \cdot 0^{3 (\frac{2}{3})} - 2 \cdot 0$

Step 2.2.3.1.3

Cancel the common factor of $3$ .

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Step 2.2.3.1.3.1

Cancel the common factor.

$y = 3 \cdot 0^{3 (\frac{2}{3})} - 2 \cdot 0$

Step 2.2.3.1.3.2

Rewrite the expression.

$y = 3 \cdot 0^{2} - 2 \cdot 0$

Step 2.2.3.1.4

Raising $0$ to any positive power yields $0$ .

$y = 3 \cdot 0 - 2 \cdot 0$

Step 2.2.3.1.5

Multiply $3$ by $0$ .

$y = 0 - 2 \cdot 0$

Step 2.2.3.1.6

Multiply $- 2$ by $0$ .

$y = 0 + 0$

Step 2.2.3.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\frac{27}{8}, 0)$

y-intercept(s): $(0, 0)$

Step 4