Calculus Examples

$f (x) = 10 x^{2} - 10 \sin (2 x)$

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

By the Sum Rule, the derivative of $10 x^{2} - 10 \sin (2 x)$ with respect to $x$ is $\frac{d}{d x} [10 x^{2}] + \frac{d}{d x} [- 10 \sin (2 x)]$ .

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

Step 1.1.1.2

Evaluate $\frac{d}{d x} [10 x^{2}]$ .

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Step 1.1.1.2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{2}$ with respect to $x$ is $10 \frac{d}{d x} [x^{2}]$ .

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

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 $n = 2$ .

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

Step 1.1.1.2.3

Multiply $2$ by $10$ .

$f' (x) = 20 x + \frac{d}{d x} (- 10 \sin (2 x))$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [- 10 \sin (2 x)]$ .

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Step 1.1.1.3.1

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

$f' (x) = 20 x - 10 \frac{d}{d x} \sin (2 x)$

Step 1.1.1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 1.1.1.3.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$f' (x) = 20 x - 10 (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (2 x))$

Step 1.1.1.3.2.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$f' (x) = 20 x - 10 (\cos (u) \frac{d}{d x} (2 x))$

Step 1.1.1.3.2.3

Replace all occurrences of $u$ with $2 x$ .

$f' (x) = 20 x - 10 (\cos (2 x) \frac{d}{d x} (2 x))$

Step 1.1.1.3.3

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

$f' (x) = 20 x - 10 (\cos (2 x) (2 \frac{d}{d x} (x)))$

Step 1.1.1.3.4

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

$f' (x) = 20 x - 10 (\cos (2 x) (2 \cdot 1))$

Step 1.1.1.3.5

Multiply $2$ by $1$ .

$f' (x) = 20 x - 10 (\cos (2 x) \cdot 2)$

Step 1.1.1.3.6

Move $2$ to the left of $\cos (2 x)$ .

$f' (x) = 20 x - 10 (2 \cdot \cos (2 x))$

Step 1.1.1.3.7

Multiply $2$ by $- 10$ .

$f' (x) = 20 x - 20 \cos (2 x)$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $20 x - 20 \cos (2 x)$ with respect to $x$ is $\frac{d}{d x} [20 x] + \frac{d}{d x} [- 20 \cos (2 x)]$ .

$f'' (x) = \frac{d}{d x} (20 x) + \frac{d}{d x} (- 20 \cos (2 x))$

Step 1.1.2.2

Evaluate $\frac{d}{d x} [20 x]$ .

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Step 1.1.2.2.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x$ with respect to $x$ is $20 \frac{d}{d x} [x]$ .

$f'' (x) = 20 \frac{d}{d x} (x) + \frac{d}{d x} (- 20 \cos (2 x))$

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 $n = 1$ .

$f'' (x) = 20 \cdot 1 + \frac{d}{d x} (- 20 \cos (2 x))$

Step 1.1.2.2.3

Multiply $20$ by $1$ .

$f'' (x) = 20 + \frac{d}{d x} (- 20 \cos (2 x))$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [- 20 \cos (2 x)]$ .

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Step 1.1.2.3.1

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20 \cos (2 x)$ with respect to $x$ is $- 20 \frac{d}{d x} [\cos (2 x)]$ .

$f'' (x) = 20 - 20 \frac{d}{d x} \cos (2 x)$

Step 1.1.2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 2 x$ .

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Step 1.1.2.3.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$f'' (x) = 20 - 20 (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (2 x))$

Step 1.1.2.3.2.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$f'' (x) = 20 - 20 (- \sin (u) \frac{d}{d x} (2 x))$

Step 1.1.2.3.2.3

Replace all occurrences of $u$ with $2 x$ .

$f'' (x) = 20 - 20 (- \sin (2 x) \frac{d}{d x} (2 x))$

Step 1.1.2.3.3

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

$f'' (x) = 20 - 20 (- \sin (2 x) (2 \frac{d}{d x} (x)))$

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 $n = 1$ .

$f'' (x) = 20 - 20 (- \sin (2 x) (2 \cdot 1))$

Step 1.1.2.3.5

Multiply $2$ by $1$ .

$f'' (x) = 20 - 20 (- \sin (2 x) \cdot 2)$

Step 1.1.2.3.6

Multiply $2$ by $- 1$ .

$f'' (x) = 20 - 20 (- 2 \sin (2 x))$

Step 1.1.2.3.7

Multiply $- 2$ by $- 20$ .

$f'' (x) = 20 + 40 \sin (2 x)$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $20 + 40 \sin (2 x)$ .

$20 + 40 \sin (2 x)$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $20 + 40 \sin (2 x) = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$20 + 40 \sin (2 x) = 0$

Step 1.2.2

Subtract $20$ from both sides of the equation.

$40 \sin (2 x) = - 20$

Step 1.2.3

Divide each term in $40 \sin (2 x) = - 20$ by $40$ and simplify.

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Step 1.2.3.1

Divide each term in $40 \sin (2 x) = - 20$ by $40$ .

$\frac{40 \sin (2 x)}{40} = \frac{- 20}{40}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $40$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{40 \sin (2 x)}{40} = \frac{- 20}{40}$

Step 1.2.3.2.1.2

Divide $\sin (2 x)$ by $1$ .

$\sin (2 x) = \frac{- 20}{40}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Cancel the common factor of $- 20$ and $40$ .

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Step 1.2.3.3.1.1

Factor $20$ out of $- 20$ .

$\sin (2 x) = \frac{20 (- 1)}{40}$

Step 1.2.3.3.1.2

Cancel the common factors.

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Step 1.2.3.3.1.2.1

Factor $20$ out of $40$ .

$\sin (2 x) = \frac{20 \cdot - 1}{20 \cdot 2}$

Step 1.2.3.3.1.2.2

Cancel the common factor.

$\sin (2 x) = \frac{20 \cdot - 1}{20 \cdot 2}$

Step 1.2.3.3.1.2.3

Rewrite the expression.

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

Step 1.2.3.3.2

Move the negative in front of the fraction.

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

Step 1.2.4

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

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

Step 1.2.5

Simplify the right side.

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Step 1.2.5.1

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

$2 x = - \frac{π}{6}$

Step 1.2.6

Divide each term in $2 x = - \frac{π}{6}$ by $2$ and simplify.

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Step 1.2.6.1

Divide each term in $2 x = - \frac{π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{- \frac{π}{6}}{2}$

Step 1.2.6.2

Simplify the left side.

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Step 1.2.6.2.1

Cancel the common factor of $2$ .

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Step 1.2.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- \frac{π}{6}}{2}$

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{6}}{2}$

Step 1.2.6.3

Simplify the right side.

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Step 1.2.6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{6} \cdot \frac{1}{2}$

Step 1.2.6.3.2

Multiply $- \frac{π}{6} \cdot \frac{1}{2}$ .

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Step 1.2.6.3.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{6}$ .

$x = - \frac{π}{2 \cdot 6}$

Step 1.2.6.3.2.2

Multiply $2$ by $6$ .

$x = - \frac{π}{12}$

Step 1.2.7

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$2 x = 2 π + \frac{π}{6} + π$

Step 1.2.8

Simplify the expression to find the second solution.

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Step 1.2.8.1

Subtract $2 π$ from $2 π + \frac{π}{6} + π$ .

$2 x = 2 π + \frac{π}{6} + π - 2 π$

Step 1.2.8.2

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{6} + π$ .

$2 x = \frac{7 π}{6}$

Step 1.2.8.3

Divide each term in $2 x = \frac{7 π}{6}$ by $2$ and simplify.

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Step 1.2.8.3.1

Divide each term in $2 x = \frac{7 π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{7 π}{6}}{2}$

Step 1.2.8.3.2

Simplify the left side.

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Step 1.2.8.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.8.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{7 π}{6}}{2}$

Step 1.2.8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{7 π}{6}}{2}$

Step 1.2.8.3.3

Simplify the right side.

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Step 1.2.8.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 π}{6} \cdot \frac{1}{2}$

Step 1.2.8.3.3.2

Multiply $\frac{7 π}{6} \cdot \frac{1}{2}$ .

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Step 1.2.8.3.3.2.1

Multiply $\frac{7 π}{6}$ by $\frac{1}{2}$ .

$x = \frac{7 π}{6 \cdot 2}$

Step 1.2.8.3.3.2.2

Multiply $6$ by $2$ .

$x = \frac{7 π}{12}$

Step 1.2.9

Find the period of $\sin (2 x)$ .

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Step 1.2.9.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 1.2.9.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 1.2.9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 1.2.9.4

Cancel the common factor of $2$ .

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Step 1.2.9.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 1.2.9.4.2

Divide $π$ by $1$ .

$π$

Step 1.2.10

Add $π$ to every negative angle to get positive angles.

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Step 1.2.10.1

Add $π$ to $- \frac{π}{12}$ to find the positive angle.

$- \frac{π}{12} + π$

Step 1.2.10.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$π \cdot \frac{12}{12} - \frac{π}{12}$

Step 1.2.10.3

Combine fractions.

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Step 1.2.10.3.1

Combine $π$ and $\frac{12}{12}$ .

$\frac{π \cdot 12}{12} - \frac{π}{12}$

Step 1.2.10.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 12 - π}{12}$

Step 1.2.10.4

Simplify the numerator.

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Step 1.2.10.4.1

Move $12$ to the left of $π$ .

$\frac{12 \cdot π - π}{12}$

Step 1.2.10.4.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{12}$

Step 1.2.10.5

List the new angles.

$x = \frac{11 π}{12}$

Step 1.2.11

The period of the $\sin (2 x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{7 π}{12} + π n, \frac{11 π}{12} + π n$ , for any integer $n$

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.

$(- \infty, \infty)$

Step 4

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

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Step 4.1

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

$f'' (0) = 20 + 40 \sin (2 (0))$

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

Multiply $2$ by $0$ .

$f'' (0) = 20 + 40 \sin (0)$

Step 4.2.1.2

The exact value of $\sin (0)$ is $0$ .

$f'' (0) = 20 + 40 \cdot 0$

Step 4.2.1.3

Multiply $40$ by $0$ .

$f'' (0) = 20 + 0$

Step 4.2.2

Add $20$ and $0$ .

$f'' (0) = 20$

Step 4.2.3

The final answer is $20$ .

$20$

Step 4.3

The graph is concave up on the interval $(- \infty, \infty)$ because $f'' (0)$ is positive.

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

Step 5