Calculus Examples

$f (x) = 3 x^{4} - 8 x^{3} + 6 x^{2} + 2$

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

By the Sum Rule, the derivative of $3 x^{4} - 8 x^{3} + 6 x^{2} + 2$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [2]$ .

$f' (x) = \frac{d}{d x} (3 x^{4}) + \frac{d}{d x} (- 8 x^{3}) + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (2)$

Step 1.1.2

Evaluate $\frac{d}{d x} [3 x^{4}]$ .

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Step 1.1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$f' (x) = 3 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 8 x^{3}) + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (2)$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f' (x) = 3 (4 x^{3}) + \frac{d}{d x} (- 8 x^{3}) + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (2)$

Step 1.1.2.3

Multiply $4$ by $3$ .

$f' (x) = 12 x^{3} + \frac{d}{d x} (- 8 x^{3}) + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (2)$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 8 x^{3}]$ .

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Step 1.1.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 12 x^{3} - 8 \frac{d}{d x} x^{3} + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (2)$

Step 1.1.3.2

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

$f' (x) = 12 x^{3} - 8 (3 x^{2}) + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (2)$

Step 1.1.3.3

Multiply $3$ by $- 8$ .

$f' (x) = 12 x^{3} - 24 x^{2} + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (2)$

Step 1.1.4

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 1.1.4.1

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

$f' (x) = 12 x^{3} - 24 x^{2} + 6 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (2)$

Step 1.1.4.2

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

$f' (x) = 12 x^{3} - 24 x^{2} + 6 (2 x) + \frac{d}{d x} (2)$

Step 1.1.4.3

Multiply $2$ by $6$ .

$f' (x) = 12 x^{3} - 24 x^{2} + 12 x + \frac{d}{d x} (2)$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f' (x) = 12 x^{3} - 24 x^{2} + 12 x + 0$

Step 1.1.5.2

Add $12 x^{3} - 24 x^{2} + 12 x$ and $0$ .

$f' (x) = 12 x^{3} - 24 x^{2} + 12 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{3} - 24 x^{2} + 12 x$ .

$12 x^{3} - 24 x^{2} + 12 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $12 x^{3} - 24 x^{2} + 12 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$12 x^{3} - 24 x^{2} + 12 x = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $12 x$ out of $12 x^{3} - 24 x^{2} + 12 x$ .

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Step 2.2.1.1

Factor $12 x$ out of $12 x^{3}$ .

$12 x (x^{2}) - 24 x^{2} + 12 x = 0$

Step 2.2.1.2

Factor $12 x$ out of $- 24 x^{2}$ .

$12 x (x^{2}) + 12 x (- 2 x) + 12 x = 0$

Step 2.2.1.3

Factor $12 x$ out of $12 x$ .

$12 x (x^{2}) + 12 x (- 2 x) + 12 x (1) = 0$

Step 2.2.1.4

Factor $12 x$ out of $12 x (x^{2}) + 12 x (- 2 x)$ .

$12 x (x^{2} - 2 x) + 12 x (1) = 0$

Step 2.2.1.5

Factor $12 x$ out of $12 x (x^{2} - 2 x) + 12 x (1)$ .

$12 x (x^{2} - 2 x + 1) = 0$

Step 2.2.2

Factor using the perfect square rule.

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Step 2.2.2.1

Rewrite $1$ as $1^{2}$ .

$12 x (x^{2} - 2 x + 1^{2}) = 0$

Step 2.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 2.2.2.3

Rewrite the polynomial.

$12 x (x^{2} - 2 \cdot x \cdot 1 + 1^{2}) = 0$

Step 2.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$12 x {(x - 1)}^{2} = 0$

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 $0$ .

$x = 0$

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

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set ${(x - 1)}^{2}$ equal to $0$ .

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

Step 2.5.2

Solve ${(x - 1)}^{2} = 0$ for $x$ .

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Step 2.5.2.1

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

$x - 1 = 0$

Step 2.5.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.6

The final solution is all the values that make $12 x {(x - 1)}^{2} = 0$ true.

$x = 0, 1$

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 $3 x^{4} - 8 x^{3} + 6 x^{2} + 2$ 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 $x$ .

$3 {(0)}^{4} - 8 {(0)}^{3} + 6 {(0)}^{2} + 2$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$3 \cdot 0 - 8 {(0)}^{3} + 6 {(0)}^{2} + 2$

Step 4.1.2.1.2

Multiply $3$ by $0$ .

$0 - 8 {(0)}^{3} + 6 {(0)}^{2} + 2$

Step 4.1.2.1.3

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

$0 - 8 \cdot 0 + 6 {(0)}^{2} + 2$

Step 4.1.2.1.4

Multiply $- 8$ by $0$ .

$0 + 0 + 6 {(0)}^{2} + 2$

Step 4.1.2.1.5

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

$0 + 0 + 6 \cdot 0 + 2$

Step 4.1.2.1.6

Multiply $6$ by $0$ .

$0 + 0 + 0 + 2$

Step 4.1.2.2

Simplify by adding numbers.

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Step 4.1.2.2.1

Add $0$ and $0$ .

$0 + 0 + 2$

Step 4.1.2.2.2

Add $0$ and $0$ .

$0 + 2$

Step 4.1.2.2.3

Add $0$ and $2$ .

$2$

Step 4.2

Evaluate at $x = 1$ .

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Step 4.2.1

Substitute $1$ for $x$ .

$3 {(1)}^{4} - 8 {(1)}^{3} + 6 {(1)}^{2} + 2$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

One to any power is one.

$3 \cdot 1 - 8 {(1)}^{3} + 6 {(1)}^{2} + 2$

Step 4.2.2.1.2

Multiply $3$ by $1$ .

$3 - 8 {(1)}^{3} + 6 {(1)}^{2} + 2$

Step 4.2.2.1.3

One to any power is one.

$3 - 8 \cdot 1 + 6 {(1)}^{2} + 2$

Step 4.2.2.1.4

Multiply $- 8$ by $1$ .

$3 - 8 + 6 {(1)}^{2} + 2$

Step 4.2.2.1.5

One to any power is one.

$3 - 8 + 6 \cdot 1 + 2$

Step 4.2.2.1.6

Multiply $6$ by $1$ .

$3 - 8 + 6 + 2$

Step 4.2.2.2

Simplify by adding and subtracting.

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Step 4.2.2.2.1

Subtract $8$ from $3$ .

$- 5 + 6 + 2$

Step 4.2.2.2.2

Add $- 5$ and $6$ .

$1 + 2$

Step 4.2.2.2.3

Add $1$ and $2$ .

$3$

Step 4.3

List all of the points.

$(0, 2), (1, 3)$

Step 5