Calculus Examples

$f (x) = x^{3} - \frac{5}{2} x^{2} + 4 x - 2$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{3} - \frac{5}{2} x^{2} + 4 x - 2$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{5}{2} x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{5}{2} x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.1.2

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

$3 x^{2} + \frac{d}{d x} [- \frac{5}{2} x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{5}{2} x^{2}]$ .

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Step 1.2.1

Since $- \frac{5}{2}$ is constant with respect to $x$ , the derivative of $- \frac{5}{2} x^{2}$ with respect to $x$ is $- \frac{5}{2} \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - \frac{5}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.2.2

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

$3 x^{2} - \frac{5}{2} (2 x) + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.2.3

Multiply $2$ by $- 1$ .

$3 x^{2} - 2 (\frac{5}{2}) x + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.2.4

Combine $- 2$ and $\frac{5}{2}$ .

$3 x^{2} + \frac{- 2 \cdot 5}{2} x + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.2.5

Multiply $- 2$ by $5$ .

$3 x^{2} + \frac{- 10}{2} x + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.2.6

Combine $\frac{- 10}{2}$ and $x$ .

$3 x^{2} + \frac{- 10 x}{2} + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.2.7

Cancel the common factor of $- 10$ and $2$ .

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Step 1.2.7.1

Factor $2$ out of $- 10 x$ .

$3 x^{2} + \frac{2 (- 5 x)}{2} + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.2.7.2

Cancel the common factors.

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Step 1.2.7.2.1

Factor $2$ out of $2$ .

$3 x^{2} + \frac{2 (- 5 x)}{2 (1)} + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.2.7.2.2

Cancel the common factor.

$3 x^{2} + \frac{2 (- 5 x)}{2 \cdot 1} + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.2.7.2.3

Rewrite the expression.

$3 x^{2} + \frac{- 5 x}{1} + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.2.7.2.4

Divide $- 5 x$ by $1$ .

$3 x^{2} - 5 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 1.3

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

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Step 1.3.1

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

$3 x^{2} - 5 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 1.3.2

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

$3 x^{2} - 5 x + 4 \cdot 1 + \frac{d}{d x} [- 2]$

Step 1.3.3

Multiply $4$ by $1$ .

$3 x^{2} - 5 x + 4 + \frac{d}{d x} [- 2]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

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

$3 x^{2} - 5 x + 4 + 0$

Step 1.4.2

Add $3 x^{2} - 5 x + 4$ and $0$ .

$3 x^{2} - 5 x + 4$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $3 x^{2} - 5 x + 4$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [4]$ .

$f'' (x) = \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (4)$

Step 2.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (4)$

Step 2.2.2

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

$f'' (x) = 3 (2 x) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (4)$

Step 2.2.3

Multiply $2$ by $3$ .

$f'' (x) = 6 x + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (4)$

Step 2.3

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

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Step 2.3.1

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

$f'' (x) = 6 x - 5 \frac{d}{d x} x + \frac{d}{d x} (4)$

Step 2.3.2

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

$f'' (x) = 6 x - 5 \cdot 1 + \frac{d}{d x} (4)$

Step 2.3.3

Multiply $- 5$ by $1$ .

$f'' (x) = 6 x - 5 + \frac{d}{d x} (4)$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f'' (x) = 6 x - 5 + 0$

Step 2.4.2

Add $6 x - 5$ and $0$ .

$f'' (x) = 6 x - 5$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$3 x^{2} - 5 x + 4 = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6