Calculus Examples

$f (x) = 10 x^{3} {(x - 1)}^{2}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

$f' (x) = \frac{d}{d x} (10 x^{3} ((x - 1) (x - 1)))$

Step 1.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$f' (x) = \frac{d}{d x} (10 x^{3} (x (x - 1) - 1 (x - 1)))$

Step 1.1.2.2

Apply the distributive property.

$f' (x) = \frac{d}{d x} (10 x^{3} (x \cdot x + x \cdot - 1 - 1 (x - 1)))$

Step 1.1.2.3

Apply the distributive property.

$f' (x) = \frac{d}{d x} (10 x^{3} (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1))$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

$f' (x) = \frac{d}{d x} (10 x^{3} (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1))$

Step 1.1.3.1.2

Move $- 1$ to the left of $x$ .

$f' (x) = \frac{d}{d x} (10 x^{3} (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1))$

Step 1.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

$f' (x) = \frac{d}{d x} (10 x^{3} (x^{2} - x - 1 x - 1 \cdot - 1))$

Step 1.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$f' (x) = \frac{d}{d x} (10 x^{3} (x^{2} - x - x - 1 \cdot - 1))$

Step 1.1.3.1.5

Multiply $- 1$ by $- 1$ .

$f' (x) = \frac{d}{d x} (10 x^{3} (x^{2} - x - x + 1))$

Step 1.1.3.2

Subtract $x$ from $- x$ .

$f' (x) = \frac{d}{d x} (10 x^{3} (x^{2} - 2 x + 1))$

Step 1.1.4

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{3} (x^{2} - 2 x + 1)$ with respect to $x$ is $10 \frac{d}{d x} [x^{3} (x^{2} - 2 x + 1)]$ .

$f' (x) = 10 \frac{d}{d x} (x^{3} (x^{2} - 2 x + 1))$

Step 1.1.5

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 $f (x) = x^{3}$ and $g (x) = x^{2} - 2 x + 1$ .

$f' (x) = 10 (x^{3} \frac{d}{d x} (x^{2} - 2 x + 1) + (x^{2} - 2 x + 1) \frac{d}{d x} (x^{3}))$

Step 1.1.6

Differentiate.

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Step 1.1.6.1

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

$f' (x) = 10 (x^{3} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 2 x) + \frac{d}{d x} (1)) + (x^{2} - 2 x + 1) \frac{d}{d x} (x^{3}))$

Step 1.1.6.2

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

$f' (x) = 10 (x^{3} (2 x + \frac{d}{d x} (- 2 x) + \frac{d}{d x} (1)) + (x^{2} - 2 x + 1) \frac{d}{d x} (x^{3}))$

Step 1.1.6.3

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

$f' (x) = 10 (x^{3} (2 x - 2 \frac{d}{d x} x + \frac{d}{d x} (1)) + (x^{2} - 2 x + 1) \frac{d}{d x} (x^{3}))$

Step 1.1.6.4

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

$f' (x) = 10 (x^{3} (2 x - 2 \cdot 1 + \frac{d}{d x} (1)) + (x^{2} - 2 x + 1) \frac{d}{d x} (x^{3}))$

Step 1.1.6.5

Multiply $- 2$ by $1$ .

$f' (x) = 10 (x^{3} (2 x - 2 + \frac{d}{d x} (1)) + (x^{2} - 2 x + 1) \frac{d}{d x} (x^{3}))$

Step 1.1.6.6

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

$f' (x) = 10 (x^{3} (2 x - 2 + 0) + (x^{2} - 2 x + 1) \frac{d}{d x} (x^{3}))$

Step 1.1.6.7

Add $2 x - 2$ and $0$ .

$f' (x) = 10 (x^{3} (2 x - 2) + (x^{2} - 2 x + 1) \frac{d}{d x} (x^{3}))$

Step 1.1.6.8

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

$f' (x) = 10 (x^{3} (2 x - 2) + (x^{2} - 2 x + 1) (3 x^{2}))$

Step 1.1.6.9

Move $3$ to the left of $x^{2} - 2 x + 1$ .

$f' (x) = 10 (x^{3} (2 x - 2) + 3 \cdot ((x^{2} - 2 x + 1) x^{2}))$

Step 1.1.7

Simplify.

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Step 1.1.7.1

Apply the distributive property.

$f' (x) = 10 (x^{3} (2 x) + x^{3} \cdot - 2 + 3 (x^{2} - 2 x + 1) x^{2})$

Step 1.1.7.2

Apply the distributive property.

$f' (x) = 10 (x^{3} (2 x) + x^{3} \cdot - 2 + (3 x^{2} + 3 (- 2 x) + 3 \cdot 1) x^{2})$

Step 1.1.7.3

Apply the distributive property.

$f' (x) = 10 (x^{3} (2 x) + x^{3} \cdot - 2 + 3 x^{2} x^{2} + 3 (- 2 x) x^{2} + 3 \cdot (1 x^{2}))$

Step 1.1.7.4

Apply the distributive property.

$f' (x) = 10 (x^{3} (2 x)) + 10 (x^{3} \cdot - 2) + 10 (3 x^{2} x^{2}) + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5

Combine terms.

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Step 1.1.7.5.1

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

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Step 1.1.7.5.1.1

Move $x$ .

$f' (x) = 10 (x \cdot x^{3} \cdot 2) + 10 (x^{3} \cdot - 2) + 10 (3 x^{2} x^{2}) + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.1.2

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

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Step 1.1.7.5.1.2.1

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

$f' (x) = 10 (x \cdot x^{3} \cdot 2) + 10 (x^{3} \cdot - 2) + 10 (3 x^{2} x^{2}) + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.1.2.2

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

$f' (x) = 10 (x^{1 + 3} \cdot 2) + 10 (x^{3} \cdot - 2) + 10 (3 x^{2} x^{2}) + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.1.3

Add $1$ and $3$ .

$f' (x) = 10 (x^{4} \cdot 2) + 10 (x^{3} \cdot - 2) + 10 (3 x^{2} x^{2}) + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.2

Move $2$ to the left of $x^{4}$ .

$f' (x) = 10 (2 \cdot x^{4}) + 10 (x^{3} \cdot - 2) + 10 (3 x^{2} x^{2}) + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.3

Multiply $2$ by $10$ .

$f' (x) = 20 x^{4} + 10 (x^{3} \cdot - 2) + 10 (3 x^{2} x^{2}) + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.4

Move $- 2$ to the left of $x^{3}$ .

$f' (x) = 20 x^{4} + 10 (- 2 \cdot x^{3}) + 10 (3 x^{2} x^{2}) + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.5

Multiply $- 2$ by $10$ .

$f' (x) = 20 x^{4} - 20 x^{3} + 10 (3 x^{2} x^{2}) + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.6

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

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Step 1.1.7.5.6.1

Move $x^{2}$ .

$f' (x) = 20 x^{4} - 20 x^{3} + 10 (3 (x^{2} x^{2})) + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.6.2

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

$f' (x) = 20 x^{4} - 20 x^{3} + 10 (3 x^{2 + 2}) + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.6.3

Add $2$ and $2$ .

$f' (x) = 20 x^{4} - 20 x^{3} + 10 (3 x^{4}) + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.7

Multiply $3$ by $10$ .

$f' (x) = 20 x^{4} - 20 x^{3} + 30 x^{4} + 10 (3 (- 2 x) x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.8

Multiply $- 2$ by $3$ .

$f' (x) = 20 x^{4} - 20 x^{3} + 30 x^{4} + 10 (- 6 x \cdot x^{2}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.9

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

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Step 1.1.7.5.9.1

Move $x^{2}$ .

$f' (x) = 20 x^{4} - 20 x^{3} + 30 x^{4} + 10 (- 6 (x^{2} x)) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.9.2

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

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Step 1.1.7.5.9.2.1

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

$f' (x) = 20 x^{4} - 20 x^{3} + 30 x^{4} + 10 (- 6 (x^{2} x)) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.9.2.2

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

$f' (x) = 20 x^{4} - 20 x^{3} + 30 x^{4} + 10 (- 6 x^{2 + 1}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.9.3

Add $2$ and $1$ .

$f' (x) = 20 x^{4} - 20 x^{3} + 30 x^{4} + 10 (- 6 x^{3}) + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.10

Multiply $- 6$ by $10$ .

$f' (x) = 20 x^{4} - 20 x^{3} + 30 x^{4} - 60 x^{3} + 10 (3 \cdot (1 x^{2}))$

Step 1.1.7.5.11

Multiply $3$ by $1$ .

$f' (x) = 20 x^{4} - 20 x^{3} + 30 x^{4} - 60 x^{3} + 10 (3 x^{2})$

Step 1.1.7.5.12

Multiply $3$ by $10$ .

$f' (x) = 20 x^{4} - 20 x^{3} + 30 x^{4} - 60 x^{3} + 30 x^{2}$

Step 1.1.7.5.13

Add $20 x^{4}$ and $30 x^{4}$ .

$f' (x) = 50 x^{4} - 20 x^{3} - 60 x^{3} + 30 x^{2}$

Step 1.1.7.5.14

Subtract $60 x^{3}$ from $- 20 x^{3}$ .

$f' (x) = 50 x^{4} - 80 x^{3} + 30 x^{2}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $50 x^{4} - 80 x^{3} + 30 x^{2}$ .

$50 x^{4} - 80 x^{3} + 30 x^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $50 x^{4} - 80 x^{3} + 30 x^{2} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$50 x^{4} - 80 x^{3} + 30 x^{2} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $10 x^{2}$ out of $50 x^{4} - 80 x^{3} + 30 x^{2}$ .

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Step 2.2.1.1

Factor $10 x^{2}$ out of $50 x^{4}$ .

$10 x^{2} (5 x^{2}) - 80 x^{3} + 30 x^{2} = 0$

Step 2.2.1.2

Factor $10 x^{2}$ out of $- 80 x^{3}$ .

$10 x^{2} (5 x^{2}) + 10 x^{2} (- 8 x) + 30 x^{2} = 0$

Step 2.2.1.3

Factor $10 x^{2}$ out of $30 x^{2}$ .

$10 x^{2} (5 x^{2}) + 10 x^{2} (- 8 x) + 10 x^{2} (3) = 0$

Step 2.2.1.4

Factor $10 x^{2}$ out of $10 x^{2} (5 x^{2}) + 10 x^{2} (- 8 x)$ .

$10 x^{2} (5 x^{2} - 8 x) + 10 x^{2} (3) = 0$

Step 2.2.1.5

Factor $10 x^{2}$ out of $10 x^{2} (5 x^{2} - 8 x) + 10 x^{2} (3)$ .

$10 x^{2} (5 x^{2} - 8 x + 3) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor by grouping.

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Step 2.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot 3 = 15$ and whose sum is $b = - 8$ .

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Step 2.2.2.1.1.1

Factor $- 8$ out of $- 8 x$ .

$10 x^{2} (5 x^{2} - 8 x + 3) = 0$

Step 2.2.2.1.1.2

Rewrite $- 8$ as $- 3$ plus $- 5$

$10 x^{2} (5 x^{2} + (- 3 - 5) x + 3) = 0$

Step 2.2.2.1.1.3

Apply the distributive property.

$10 x^{2} (5 x^{2} - 3 x - 5 x + 3) = 0$

Step 2.2.2.1.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1.2.1

Group the first two terms and the last two terms.

$10 x^{2} ((5 x^{2} - 3 x) - 5 x + 3) = 0$

Step 2.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$10 x^{2} (x (5 x - 3) - (5 x - 3)) = 0$

Step 2.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 x - 3$ .

$10 x^{2} ((5 x - 3) (x - 1)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$10 x^{2} (5 x - 3) (x - 1) = 0$

Step 2.3

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

$x^{2} = 0$

$5 x - 3 = 0$

$x - 1 = 0$

Step 2.4

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

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Step 2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.4.2.1

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

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.4.2.2.1

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

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

Step 2.4.2.2.2

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

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

Set $5 x - 3$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $5 x - 3$ equal to $0$ .

$5 x - 3 = 0$

Step 2.5.2

Solve $5 x - 3 = 0$ for $x$ .

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Step 2.5.2.1

Add $3$ to both sides of the equation.

$5 x = 3$

Step 2.5.2.2

Divide each term in $5 x = 3$ by $5$ and simplify.

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Step 2.5.2.2.1

Divide each term in $5 x = 3$ by $5$ .

$\frac{5 x}{5} = \frac{3}{5}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{3}{5}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{5}$

Step 2.6

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.7

The final solution is all the values that make $10 x^{2} (5 x - 3) (x - 1) = 0$ true.

$x = 0, \frac{3}{5}, 1$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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.

Step 4

Evaluate $10 x^{3} {(x - 1)}^{2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$10 {(0)}^{3} {((0) - 1)}^{2}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$10 \cdot 0 {((0) - 1)}^{2}$

Step 4.1.2.2

Multiply $10$ by $0$ .

$0 {((0) - 1)}^{2}$

Step 4.1.2.3

Subtract $1$ from $0$ .

$0 {(- 1)}^{2}$

Step 4.1.2.4

Raise $- 1$ to the power of $2$ .

$0 \cdot 1$

Step 4.1.2.5

Multiply $0$ by $1$ .

$0$

Step 4.2

Evaluate at $x = \frac{3}{5}$ .

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Step 4.2.1

Substitute $\frac{3}{5}$ for $x$ .

$10 {(\frac{3}{5})}^{3} {((\frac{3}{5}) - 1)}^{2}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the expression.

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Step 4.2.2.1.1

Apply the product rule to $\frac{3}{5}$ .

$10 \frac{3^{3}}{5^{3}} {((\frac{3}{5}) - 1)}^{2}$

Step 4.2.2.1.2

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

$10 \frac{27}{5^{3}} {((\frac{3}{5}) - 1)}^{2}$

Step 4.2.2.1.3

Raise $5$ to the power of $3$ .

$10 (\frac{27}{125}) {((\frac{3}{5}) - 1)}^{2}$

Step 4.2.2.2

Cancel the common factor of $5$ .

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Step 4.2.2.2.1

Factor $5$ out of $10$ .

$5 (2) \frac{27}{125} {((\frac{3}{5}) - 1)}^{2}$

Step 4.2.2.2.2

Factor $5$ out of $125$ .

$5 \cdot 2 \frac{27}{5 \cdot 25} {((\frac{3}{5}) - 1)}^{2}$

Step 4.2.2.2.3

Cancel the common factor.

$5 \cdot 2 \frac{27}{5 \cdot 25} {((\frac{3}{5}) - 1)}^{2}$

Step 4.2.2.2.4

Rewrite the expression.

$2 (\frac{27}{25}) {((\frac{3}{5}) - 1)}^{2}$

Step 4.2.2.3

Combine $2$ and $\frac{27}{25}$ .

$\frac{2 \cdot 27}{25} {((\frac{3}{5}) - 1)}^{2}$

Step 4.2.2.4

Multiply $2$ by $27$ .

$\frac{54}{25} {((\frac{3}{5}) - 1)}^{2}$

Step 4.2.2.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{54}{25} {(\frac{3}{5} - 1 \cdot \frac{5}{5})}^{2}$

Step 4.2.2.6

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{54}{25} {(\frac{3}{5} + \frac{- 1 \cdot 5}{5})}^{2}$

Step 4.2.2.7

Combine the numerators over the common denominator.

$\frac{54}{25} {(\frac{3 - 1 \cdot 5}{5})}^{2}$

Step 4.2.2.8

Simplify the numerator.

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Step 4.2.2.8.1

Multiply $- 1$ by $5$ .

$\frac{54}{25} {(\frac{3 - 5}{5})}^{2}$

Step 4.2.2.8.2

Subtract $5$ from $3$ .

$\frac{54}{25} {(\frac{- 2}{5})}^{2}$

Step 4.2.2.9

Move the negative in front of the fraction.

$\frac{54}{25} {(- \frac{2}{5})}^{2}$

Step 4.2.2.10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.2.10.1

Apply the product rule to $- \frac{2}{5}$ .

$\frac{54}{25} ({(- 1)}^{2} {(\frac{2}{5})}^{2})$

Step 4.2.2.10.2

Apply the product rule to $\frac{2}{5}$ .

$\frac{54}{25} ({(- 1)}^{2} \frac{2^{2}}{5^{2}})$

Step 4.2.2.11

Combine fractions.

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Step 4.2.2.11.1

Raise $- 1$ to the power of $2$ .

$\frac{54}{25} (1 \frac{2^{2}}{5^{2}})$

Step 4.2.2.11.2

Multiply $\frac{2^{2}}{5^{2}}$ by $1$ .

$\frac{54}{25} \cdot \frac{2^{2}}{5^{2}}$

Step 4.2.2.11.3

Combine.

$\frac{54 \cdot 2^{2}}{25 \cdot 5^{2}}$

Step 4.2.2.11.4

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

$\frac{54 \cdot 4}{25 \cdot 5^{2}}$

Step 4.2.2.12

Simplify the denominator.

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Step 4.2.2.12.1

Rewrite $25$ as $5^{2}$ .

$\frac{54 \cdot 4}{5^{2} \cdot 5^{2}}$

Step 4.2.2.12.2

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

$\frac{54 \cdot 4}{5^{2 + 2}}$

Step 4.2.2.12.3

Add $2$ and $2$ .

$\frac{54 \cdot 4}{5^{4}}$

Step 4.2.2.13

Simplify the expression.

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Step 4.2.2.13.1

Multiply $54$ by $4$ .

$\frac{216}{5^{4}}$

Step 4.2.2.13.2

Raise $5$ to the power of $4$ .

$\frac{216}{625}$

Step 4.3

Evaluate at $x = 1$ .

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Step 4.3.1

Substitute $1$ for $x$ .

$10 {(1)}^{3} {((1) - 1)}^{2}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

One to any power is one.

$10 \cdot 1 {((1) - 1)}^{2}$

Step 4.3.2.2

Multiply $10$ by $1$ .

$10 {((1) - 1)}^{2}$

Step 4.3.2.3

Subtract $1$ from $1$ .

$10 \cdot 0^{2}$

Step 4.3.2.4

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

$10 \cdot 0$

Step 4.3.2.5

Multiply $10$ by $0$ .

$0$

Step 4.4

List all of the points.

$(0, 0), (\frac{3}{5}, \frac{216}{625}), (1, 0)$

Step 5