Calculus Examples

$f (x) = 3 x^{4} - 4 x^{3} - 6 x^{2} + 12 x + 1$

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

By the Sum Rule, the derivative of $3 x^{4} - 4 x^{3} - 6 x^{2} + 12 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [1]$ .

$f' (x) = \frac{d}{d x} (3 x^{4}) + \frac{d}{d x} (- 4 x^{3}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (1)$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$f' (x) = 3 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 4 x^{3}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (1)$

Step 1.1.1.2.2

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

$f' (x) = 3 (4 x^{3}) + \frac{d}{d x} (- 4 x^{3}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (1)$

Step 1.1.1.2.3

Multiply $4$ by $3$ .

$f' (x) = 12 x^{3} + \frac{d}{d x} (- 4 x^{3}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (1)$

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$f' (x) = 12 x^{3} - 4 \frac{d}{d x} x^{3} + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (1)$

Step 1.1.1.3.2

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

$f' (x) = 12 x^{3} - 4 (3 x^{2}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (1)$

Step 1.1.1.3.3

Multiply $3$ by $- 4$ .

$f' (x) = 12 x^{3} - 12 x^{2} + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (1)$

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

$f' (x) = 12 x^{3} - 12 x^{2} - 6 \frac{d}{d x} x^{2} + \frac{d}{d x} (12 x) + \frac{d}{d x} (1)$

Step 1.1.1.4.2

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

$f' (x) = 12 x^{3} - 12 x^{2} - 6 (2 x) + \frac{d}{d x} (12 x) + \frac{d}{d x} (1)$

Step 1.1.1.4.3

Multiply $2$ by $- 6$ .

$f' (x) = 12 x^{3} - 12 x^{2} - 12 x + \frac{d}{d x} (12 x) + \frac{d}{d x} (1)$

Step 1.1.1.5

Evaluate $\frac{d}{d x} [12 x]$ .

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Step 1.1.1.5.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$f' (x) = 12 x^{3} - 12 x^{2} - 12 x + 12 \frac{d}{d x} (x) + \frac{d}{d x} (1)$

Step 1.1.1.5.2

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

$f' (x) = 12 x^{3} - 12 x^{2} - 12 x + 12 \cdot 1 + \frac{d}{d x} (1)$

Step 1.1.1.5.3

Multiply $12$ by $1$ .

$f' (x) = 12 x^{3} - 12 x^{2} - 12 x + 12 + \frac{d}{d x} (1)$

Step 1.1.1.6

Differentiate using the Constant Rule.

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Step 1.1.1.6.1

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

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

Step 1.1.1.6.2

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

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $12 x^{3} - 12 x^{2} - 12 x + 12$ with respect to $x$ is $\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [12]$ .

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

Step 1.1.2.2

Evaluate $\frac{d}{d x} [12 x^{3}]$ .

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Step 1.1.2.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{3}$ with respect to $x$ is $12 \frac{d}{d x} [x^{3}]$ .

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

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 $n = 3$ .

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

Step 1.1.2.2.3

Multiply $3$ by $12$ .

$f'' (x) = 36 x^{2} + \frac{d}{d x} (- 12 x^{2}) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (12)$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [- 12 x^{2}]$ .

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Step 1.1.2.3.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{2}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 36 x^{2} - 12 \frac{d}{d x} x^{2} + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (12)$

Step 1.1.2.3.2

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

$f'' (x) = 36 x^{2} - 12 (2 x) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (12)$

Step 1.1.2.3.3

Multiply $2$ by $- 12$ .

$f'' (x) = 36 x^{2} - 24 x + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (12)$

Step 1.1.2.4

Evaluate $\frac{d}{d x} [- 12 x]$ .

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Step 1.1.2.4.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$f'' (x) = 36 x^{2} - 24 x - 12 \frac{d}{d x} x + \frac{d}{d x} (12)$

Step 1.1.2.4.2

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

$f'' (x) = 36 x^{2} - 24 x - 12 \cdot 1 + \frac{d}{d x} (12)$

Step 1.1.2.4.3

Multiply $- 12$ by $1$ .

$f'' (x) = 36 x^{2} - 24 x - 12 + \frac{d}{d x} (12)$

Step 1.1.2.5

Differentiate using the Constant Rule.

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Step 1.1.2.5.1

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$f'' (x) = 36 x^{2} - 24 x - 12 + 0$

Step 1.1.2.5.2

Add $36 x^{2} - 24 x - 12$ and $0$ .

$f'' (x) = 36 x^{2} - 24 x - 12$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $36 x^{2} - 24 x - 12$ .

$36 x^{2} - 24 x - 12$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $36 x^{2} - 24 x - 12 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$36 x^{2} - 24 x - 12 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $12$ out of $36 x^{2} - 24 x - 12$ .

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Step 1.2.2.1.1

Factor $12$ out of $36 x^{2}$ .

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

Step 1.2.2.1.2

Factor $12$ out of $- 24 x$ .

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

Step 1.2.2.1.3

Factor $12$ out of $- 12$ .

$12 (3 x^{2}) + 12 (- 2 x) + 12 (- 1) = 0$

Step 1.2.2.1.4

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

$12 (3 x^{2} - 2 x) + 12 (- 1) = 0$

Step 1.2.2.1.5

Factor $12$ out of $12 (3 x^{2} - 2 x) + 12 (- 1)$ .

$12 (3 x^{2} - 2 x - 1) = 0$

Step 1.2.2.2

Factor.

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Step 1.2.2.2.1

Factor by grouping.

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Step 1.2.2.2.1.1

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 $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = - 2$ .

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Step 1.2.2.2.1.1.1

Factor $- 2$ out of $- 2 x$ .

$12 (3 x^{2} - 2 x - 1) = 0$

Step 1.2.2.2.1.1.2

Rewrite $- 2$ as $1$ plus $- 3$

$12 (3 x^{2} + (1 - 3) x - 1) = 0$

Step 1.2.2.2.1.1.3

Apply the distributive property.

$12 (3 x^{2} + 1 x - 3 x - 1) = 0$

Step 1.2.2.2.1.1.4

Multiply $x$ by $1$ .

$12 (3 x^{2} + x - 3 x - 1) = 0$

Step 1.2.2.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.2.2.1.2.1

Group the first two terms and the last two terms.

$12 ((3 x^{2} + x) - 3 x - 1) = 0$

Step 1.2.2.2.1.2.2

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

$12 (x (3 x + 1) - (3 x + 1)) = 0$

Step 1.2.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$12 ((3 x + 1) (x - 1)) = 0$

Step 1.2.2.2.2

Remove unnecessary parentheses.

$12 (3 x + 1) (x - 1) = 0$

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 $0$ .

$3 x + 1 = 0$

$x - 1 = 0$

Step 1.2.4

Set $3 x + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Step 1.2.4.2

Solve $3 x + 1 = 0$ for $x$ .

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Step 1.2.4.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 1.2.4.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 1.2.4.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

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

Step 1.2.4.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.4.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.4.2.2.3

Simplify the right side.

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Step 1.2.4.2.2.3.1

Move the negative in front of the fraction.

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

Step 1.2.5

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

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Step 1.2.5.1

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

$x - 1 = 0$

Step 1.2.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.6

The final solution is all the values that make $12 (3 x + 1) (x - 1) = 0$ true.

$x = - \frac{1}{3}, 1$

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

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - \frac{1}{3}) \cup (- \frac{1}{3}, 1) \cup (1, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - \frac{1}{3})$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

Replace the variable $x$ with $- 3$ in the expression.

$f'' (- 3) = 36 {(- 3)}^{2} - 24 \cdot - 3 - 12$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$f'' (- 3) = 36 \cdot 9 - 24 \cdot - 3 - 12$

Step 4.2.1.2

Multiply $36$ by $9$ .

$f'' (- 3) = 324 - 24 \cdot - 3 - 12$

Step 4.2.1.3

Multiply $- 24$ by $- 3$ .

$f'' (- 3) = 324 + 72 - 12$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $324$ and $72$ .

$f'' (- 3) = 396 - 12$

Step 4.2.2.2

Subtract $12$ from $396$ .

$f'' (- 3) = 384$

Step 4.2.3

The final answer is $384$ .

$384$

Step 4.3

The graph is concave up on the interval $(- \infty, - \frac{1}{3})$ because $f'' (- 3)$ is positive.

Concave up on $(- \infty, - \frac{1}{3})$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(- \frac{1}{3}, 1)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = 36 {(0)}^{2} - 24 \cdot 0 - 12$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f'' (0) = 36 \cdot 0 - 24 \cdot 0 - 12$

Step 5.2.1.2

Multiply $36$ by $0$ .

$f'' (0) = 0 - 24 \cdot 0 - 12$

Step 5.2.1.3

Multiply $- 24$ by $0$ .

$f'' (0) = 0 + 0 - 12$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $0$ and $0$ .

$f'' (0) = 0 - 12$

Step 5.2.2.2

Subtract $12$ from $0$ .

$f'' (0) = - 12$

Step 5.2.3

The final answer is $- 12$ .

$- 12$

Step 5.3

The graph is concave down on the interval $(- \frac{1}{3}, 1)$ because $f'' (0)$ is negative.

Concave down on $(- \frac{1}{3}, 1)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(1, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $4$ in the expression.

$f'' (4) = 36 {(4)}^{2} - 24 \cdot 4 - 12$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $4$ to the power of $2$ .

$f'' (4) = 36 \cdot 16 - 24 \cdot 4 - 12$

Step 6.2.1.2

Multiply $36$ by $16$ .

$f'' (4) = 576 - 24 \cdot 4 - 12$

Step 6.2.1.3

Multiply $- 24$ by $4$ .

$f'' (4) = 576 - 96 - 12$

Step 6.2.2

Simplify by subtracting numbers.

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Step 6.2.2.1

Subtract $96$ from $576$ .

$f'' (4) = 480 - 12$

Step 6.2.2.2

Subtract $12$ from $480$ .

$f'' (4) = 468$

Step 6.2.3

The final answer is $468$ .

$468$

Step 6.3

The graph is concave up on the interval $(1, \infty)$ because $f'' (4)$ is positive.

Concave up on $(1, \infty)$ since $f'' (x)$ is positive

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, - \frac{1}{3})$ since $f'' (x)$ is positive

Concave down on $(- \frac{1}{3}, 1)$ since $f'' (x)$ is negative

Concave up on $(1, \infty)$ since $f'' (x)$ is positive

Step 8