Calculus Examples

$- x^{6} + 42 x^{5} - 42 x + 19$

Step 1

Write $- x^{6} + 42 x^{5} - 42 x + 19$ as a function.

$f (x) = - x^{6} + 42 x^{5} - 42 x + 19$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $- x^{6} + 42 x^{5} - 42 x + 19$ with respect to $x$ is $\frac{d}{d x} [- x^{6}] + \frac{d}{d x} [42 x^{5}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [19]$ .

$\frac{d}{d x} [- x^{6}] + \frac{d}{d x} [42 x^{5}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [19]$

Step 2.1.2

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

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Step 2.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{6}$ with respect to $x$ is $- \frac{d}{d x} [x^{6}]$ .

$- \frac{d}{d x} [x^{6}] + \frac{d}{d x} [42 x^{5}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [19]$

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$- (6 x^{5}) + \frac{d}{d x} [42 x^{5}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [19]$

Step 2.1.2.3

Multiply $6$ by $- 1$ .

$- 6 x^{5} + \frac{d}{d x} [42 x^{5}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [19]$

Step 2.1.3

Evaluate $\frac{d}{d x} [42 x^{5}]$ .

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Step 2.1.3.1

Since $42$ is constant with respect to $x$ , the derivative of $42 x^{5}$ with respect to $x$ is $42 \frac{d}{d x} [x^{5}]$ .

$- 6 x^{5} + 42 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [19]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$- 6 x^{5} + 42 (5 x^{4}) + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [19]$

Step 2.1.3.3

Multiply $5$ by $42$ .

$- 6 x^{5} + 210 x^{4} + \frac{d}{d x} [- 42 x] + \frac{d}{d x} [19]$

Step 2.1.4

Evaluate $\frac{d}{d x} [- 42 x]$ .

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Step 2.1.4.1

Since $- 42$ is constant with respect to $x$ , the derivative of $- 42 x$ with respect to $x$ is $- 42 \frac{d}{d x} [x]$ .

$- 6 x^{5} + 210 x^{4} - 42 \frac{d}{d x} [x] + \frac{d}{d x} [19]$

Step 2.1.4.2

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

$- 6 x^{5} + 210 x^{4} - 42 \cdot 1 + \frac{d}{d x} [19]$

Step 2.1.4.3

Multiply $- 42$ by $1$ .

$- 6 x^{5} + 210 x^{4} - 42 + \frac{d}{d x} [19]$

Step 2.1.5

Differentiate using the Constant Rule.

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Step 2.1.5.1

Since $19$ is constant with respect to $x$ , the derivative of $19$ with respect to $x$ is $0$ .

$- 6 x^{5} + 210 x^{4} - 42 + 0$

Step 2.1.5.2

Add $- 6 x^{5} + 210 x^{4} - 42$ and $0$ .

$f' (x) = - 6 x^{5} + 210 x^{4} - 42$

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $- 6 x^{5} + 210 x^{4} - 42$ with respect to $x$ is $\frac{d}{d x} [- 6 x^{5}] + \frac{d}{d x} [210 x^{4}] + \frac{d}{d x} [- 42]$ .

$\frac{d}{d x} [- 6 x^{5}] + \frac{d}{d x} [210 x^{4}] + \frac{d}{d x} [- 42]$

Step 2.2.2

Evaluate $\frac{d}{d x} [- 6 x^{5}]$ .

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Step 2.2.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{5}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{5}]$ .

$- 6 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [210 x^{4}] + \frac{d}{d x} [- 42]$

Step 2.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$- 6 (5 x^{4}) + \frac{d}{d x} [210 x^{4}] + \frac{d}{d x} [- 42]$

Step 2.2.2.3

Multiply $5$ by $- 6$ .

$- 30 x^{4} + \frac{d}{d x} [210 x^{4}] + \frac{d}{d x} [- 42]$

Step 2.2.3

Evaluate $\frac{d}{d x} [210 x^{4}]$ .

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Step 2.2.3.1

Since $210$ is constant with respect to $x$ , the derivative of $210 x^{4}$ with respect to $x$ is $210 \frac{d}{d x} [x^{4}]$ .

$- 30 x^{4} + 210 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 42]$

Step 2.2.3.2

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

$- 30 x^{4} + 210 (4 x^{3}) + \frac{d}{d x} [- 42]$

Step 2.2.3.3

Multiply $4$ by $210$ .

$- 30 x^{4} + 840 x^{3} + \frac{d}{d x} [- 42]$

Step 2.2.4

Differentiate using the Constant Rule.

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Step 2.2.4.1

Since $- 42$ is constant with respect to $x$ , the derivative of $- 42$ with respect to $x$ is $0$ .

$- 30 x^{4} + 840 x^{3} + 0$

Step 2.2.4.2

Add $- 30 x^{4} + 840 x^{3}$ and $0$ .

$f'' (x) = - 30 x^{4} + 840 x^{3}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $- 30 x^{4} + 840 x^{3}$ .

$- 30 x^{4} + 840 x^{3}$

Step 3

Set the second derivative equal to $0$ then solve the equation $- 30 x^{4} + 840 x^{3} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$- 30 x^{4} + 840 x^{3} = 0$

Step 3.2

Factor $- 30 x^{3}$ out of $- 30 x^{4} + 840 x^{3}$ .

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Step 3.2.1

Factor $- 30 x^{3}$ out of $- 30 x^{4}$ .

$- 30 x^{3} x + 840 x^{3} = 0$

Step 3.2.2

Factor $- 30 x^{3}$ out of $840 x^{3}$ .

$- 30 x^{3} x - 30 x^{3} \cdot - 28 = 0$

Step 3.2.3

Factor $- 30 x^{3}$ out of $- 30 x^{3} (x) - 30 x^{3} (- 28)$ .

$- 30 x^{3} (x - 28) = 0$

Step 3.3

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

$x^{3} = 0$

$x - 28 = 0$

Step 3.4

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

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Step 3.4.1

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Step 3.4.2

Solve $x^{3} = 0$ for $x$ .

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Step 3.4.2.1

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

$x = \sqrt[3]{0}$

Step 3.4.2.2

Simplify $\sqrt[3]{0}$ .

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Step 3.4.2.2.1

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

$x = \sqrt[3]{0^{3}}$

Step 3.4.2.2.2

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

$x = 0$

Step 3.5

Set $x - 28$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 28$ equal to $0$ .

$x - 28 = 0$

Step 3.5.2

Add $28$ to both sides of the equation.

$x = 28$

Step 3.6

The final solution is all the values that make $- 30 x^{3} (x - 28) = 0$ true.

$x = 0, 28$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $0$ in $f (x) = - x^{6} + 42 x^{5} - 42 x + 19$ to find the value of $y$ .

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Step 4.1.1

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

$f (0) = - {(0)}^{6} + 42 {(0)}^{5} - 42 \cdot 0 + 19$

Step 4.1.2

Simplify the result.

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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 $0$ .

$f (0) = - 0 + 42 {(0)}^{5} - 42 \cdot 0 + 19$

Step 4.1.2.1.2

Multiply $- 1$ by $0$ .

$f (0) = 0 + 42 {(0)}^{5} - 42 \cdot 0 + 19$

Step 4.1.2.1.3

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

$f (0) = 0 + 42 \cdot 0 - 42 \cdot 0 + 19$

Step 4.1.2.1.4

Multiply $42$ by $0$ .

$f (0) = 0 + 0 - 42 \cdot 0 + 19$

Step 4.1.2.1.5

Multiply $- 42$ by $0$ .

$f (0) = 0 + 0 + 0 + 19$

Step 4.1.2.2

Simplify by adding numbers.

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Step 4.1.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 + 19$

Step 4.1.2.2.2

Add $0$ and $0$ .

$f (0) = 0 + 19$

Step 4.1.2.2.3

Add $0$ and $19$ .

$f (0) = 19$

Step 4.1.2.3

The final answer is $19$ .

$19$

Step 4.2

The point found by substituting $0$ in $f (x) = - x^{6} + 42 x^{5} - 42 x + 19$ is $(0, 19)$ . This point can be an inflection point.

$(0, 19)$

Step 4.3

Substitute $28$ in $f (x) = - x^{6} + 42 x^{5} - 42 x + 19$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $28$ in the expression.

$f (28) = - {(28)}^{6} + 42 {(28)}^{5} - 42 \cdot 28 + 19$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Raise $28$ to the power of $6$ .

$f (28) = - 1 \cdot 481890304 + 42 {(28)}^{5} - 42 \cdot 28 + 19$

Step 4.3.2.1.2

Multiply $- 1$ by $481890304$ .

$f (28) = - 481890304 + 42 {(28)}^{5} - 42 \cdot 28 + 19$

Step 4.3.2.1.3

Raise $28$ to the power of $5$ .

$f (28) = - 481890304 + 42 \cdot 17210368 - 42 \cdot 28 + 19$

Step 4.3.2.1.4

Multiply $42$ by $17210368$ .

$f (28) = - 481890304 + 722835456 - 42 \cdot 28 + 19$

Step 4.3.2.1.5

Multiply $- 42$ by $28$ .

$f (28) = - 481890304 + 722835456 - 1176 + 19$

Step 4.3.2.2

Simplify by adding and subtracting.

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Step 4.3.2.2.1

Add $- 481890304$ and $722835456$ .

$f (28) = 240945152 - 1176 + 19$

Step 4.3.2.2.2

Subtract $1176$ from $240945152$ .

$f (28) = 240943976 + 19$

Step 4.3.2.2.3

Add $240943976$ and $19$ .

$f (28) = 240943995$

Step 4.3.2.3

The final answer is $240943995$ .

$240943995$

Step 4.4

The point found by substituting $28$ in $f (x) = - x^{6} + 42 x^{5} - 42 x + 19$ is $(28, 240943995)$ . This point can be an inflection point.

$(28, 240943995)$

Step 4.5

Determine the points that could be inflection points.

$(0, 19), (28, 240943995)$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, 0) \cup (0, 28) \cup (28, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.1$ in the expression.

$f'' (- 0.1) = - 30 {(- 0.1)}^{4} + 840 {(- 0.1)}^{3}$

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 $- 0.1$ to the power of $4$ .

$f'' (- 0.1) = - 30 \cdot 0.0001 + 840 {(- 0.1)}^{3}$

Step 6.2.1.2

Multiply $- 30$ by $0.0001$ .

$f'' (- 0.1) = - 0.003 + 840 {(- 0.1)}^{3}$

Step 6.2.1.3

Raise $- 0.1$ to the power of $3$ .

$f'' (- 0.1) = - 0.003 + 840 \cdot - 0.001$

Step 6.2.1.4

Multiply $840$ by $- 0.001$ .

$f'' (- 0.1) = - 0.003 - 0.84$

Step 6.2.2

Subtract $0.84$ from $- 0.003$ .

$f'' (- 0.1) = - 0.843$

Step 6.2.3

The final answer is $- 0.843$ .

$- 0.843$

Step 6.3

At $- 0.1$ , the second derivative is $- 0.843$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, 0)$

Decreasing on $(- \infty, 0)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(0, 28)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $14$ in the expression.

$f'' (14) = - 30 {(14)}^{4} + 840 {(14)}^{3}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $14$ to the power of $4$ .

$f'' (14) = - 30 \cdot 38416 + 840 {(14)}^{3}$

Step 7.2.1.2

Multiply $- 30$ by $38416$ .

$f'' (14) = - 1152480 + 840 {(14)}^{3}$

Step 7.2.1.3

Raise $14$ to the power of $3$ .

$f'' (14) = - 1152480 + 840 \cdot 2744$

Step 7.2.1.4

Multiply $840$ by $2744$ .

$f'' (14) = - 1152480 + 2304960$

Step 7.2.2

Add $- 1152480$ and $2304960$ .

$f'' (14) = 1152480$

Step 7.2.3

The final answer is $1152480$ .

$1152480$

Step 7.3

At $14$ , the second derivative is $1152480$ . Since this is positive, the second derivative is increasing on the interval $(0, 28)$ .

Increasing on $(0, 28)$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(28, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $28.1$ in the expression.

$f'' (28.1) = - 30 {(28.1)}^{4} + 840 {(28.1)}^{3}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $28.1$ to the power of $4$ .

$f'' (28.1) = - 30 \cdot 623483.9521 + 840 {(28.1)}^{3}$

Step 8.2.1.2

Multiply $- 30$ by $623483.9521$ .

$f'' (28.1) = - 18704518.563 + 840 {(28.1)}^{3}$

Step 8.2.1.3

Raise $28.1$ to the power of $3$ .

$f'' (28.1) = - 18704518.563 + 840 \cdot 22188.041$

Step 8.2.1.4

Multiply $840$ by $22188.041$ .

$f'' (28.1) = - 18704518.563 + 18637954.44$

Step 8.2.2

Add $- 18704518.563$ and $18637954.44$ .

$f'' (28.1) = - 66564.123$

Step 8.2.3

The final answer is $- 66564.123$ .

$- 66564.123$

Step 8.3

At $28.1$ , the second derivative is $- 66564.123$ . Since this is negative, the second derivative is decreasing on the interval $(28, \infty)$

Decreasing on $(28, \infty)$ since $f'' (x) < 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(0, 19), (28, 240943995)$ .

$(0, 19), (28, 240943995)$

Step 10