Calculus Examples

$f (x) = a x^{2} + b x + c$

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 $a x^{2} + b x + c$ with respect to $x$ is $\frac{d}{d x} [a x^{2}] + \frac{d}{d x} [b x] + \frac{d}{d x} [c]$ .

$\frac{d}{d x} [a x^{2}] + \frac{d}{d x} [b x] + \frac{d}{d x} [c]$

Step 1.1.2

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

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Step 1.1.2.1

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

$a \frac{d}{d x} [x^{2}] + \frac{d}{d x} [b x] + \frac{d}{d x} [c]$

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 $n = 2$ .

$a (2 x) + \frac{d}{d x} [b x] + \frac{d}{d x} [c]$

Step 1.1.2.3

Move $2$ to the left of $a$ .

$2 a x + \frac{d}{d x} [b x] + \frac{d}{d x} [c]$

Step 1.1.3

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

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Step 1.1.3.1

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

$2 a x + b \frac{d}{d x} [x] + \frac{d}{d x} [c]$

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 $n = 1$ .

$2 a x + b \cdot 1 + \frac{d}{d x} [c]$

Step 1.1.3.3

Multiply $b$ by $1$ .

$2 a x + b + \frac{d}{d x} [c]$

Step 1.1.4

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

$2 a x + b + 0$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Add $2 a x + b$ and $0$ .

$2 a x + b$

Step 1.1.5.2

Reorder terms.

$f' (x) = b + 2 a x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $b + 2 a x$ .

$b + 2 a x$

Step 2

Set the first derivative equal to $0$ then solve the equation $b + 2 a x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$b + 2 a x = 0$

Step 2.2

Subtract $b$ from both sides of the equation.

$2 a x = - b$

Step 2.3

Divide each term in $2 a x = - b$ by $2 a$ and simplify.

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Step 2.3.1

Divide each term in $2 a x = - b$ by $2 a$ .

$\frac{2 a x}{2 a} = \frac{- b}{2 a}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 a x}{2 a} = \frac{- b}{2 a}$

Step 2.3.2.1.2

Rewrite the expression.

$\frac{a x}{a} = \frac{- b}{2 a}$

Step 2.3.2.2

Cancel the common factor of $a$ .

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Step 2.3.2.2.1

Cancel the common factor.

$\frac{a x}{a} = \frac{- b}{2 a}$

Step 2.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{- b}{2 a}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Move the negative in front of the fraction.

$x = - \frac{b}{2 a}$

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 $a x^{2} + b x + c$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - \frac{b}{2 a}$ .

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Step 4.1.1

Substitute $- \frac{b}{2 a}$ for $x$ .

$a {(- \frac{b}{2 a})}^{2} + b (- \frac{b}{2 a}) + c$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.1.1.1

Apply the product rule to $- \frac{b}{2 a}$ .

$a ({(- 1)}^{2} {(\frac{b}{2 a})}^{2}) + b (- \frac{b}{2 a}) + c$

Step 4.1.2.1.1.2

Apply the product rule to $\frac{b}{2 a}$ .

$a ({(- 1)}^{2} \frac{b^{2}}{{(2 a)}^{2}}) + b (- \frac{b}{2 a}) + c$

Step 4.1.2.1.1.3

Apply the product rule to $2 a$ .

$a ({(- 1)}^{2} \frac{b^{2}}{2^{2} a^{2}}) + b (- \frac{b}{2 a}) + c$

Step 4.1.2.1.2

Rewrite using the commutative property of multiplication.

${(- 1)}^{2} a \frac{b^{2}}{2^{2} a^{2}} + b (- \frac{b}{2 a}) + c$

Step 4.1.2.1.3

Cancel the common factor of $a$ .

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Step 4.1.2.1.3.1

Factor $a$ out of ${(- 1)}^{2} a$ .

$a {(- 1)}^{2} \frac{b^{2}}{2^{2} a^{2}} + b (- \frac{b}{2 a}) + c$

Step 4.1.2.1.3.2

Factor $a$ out of $2^{2} a^{2}$ .

$a {(- 1)}^{2} \frac{b^{2}}{a (2^{2} a)} + b (- \frac{b}{2 a}) + c$

Step 4.1.2.1.3.3

Cancel the common factor.

$a {(- 1)}^{2} \frac{b^{2}}{a (2^{2} a)} + b (- \frac{b}{2 a}) + c$

Step 4.1.2.1.3.4

Rewrite the expression.

${(- 1)}^{2} \frac{b^{2}}{2^{2} a} + b (- \frac{b}{2 a}) + c$

Step 4.1.2.1.4

Combine ${(- 1)}^{2}$ and $\frac{b^{2}}{2^{2} a}$ .

$\frac{{(- 1)}^{2} b^{2}}{2^{2} a} + b (- \frac{b}{2 a}) + c$

Step 4.1.2.1.5

Simplify the numerator.

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Step 4.1.2.1.5.1

Raise $- 1$ to the power of $2$ .

$\frac{1 b^{2}}{2^{2} a} + b (- \frac{b}{2 a}) + c$

Step 4.1.2.1.5.2

Multiply $b^{2}$ by $1$ .

$\frac{b^{2}}{2^{2} a} + b (- \frac{b}{2 a}) + c$

Step 4.1.2.1.6

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

$\frac{b^{2}}{4 a} + b (- \frac{b}{2 a}) + c$

Step 4.1.2.1.7

Rewrite using the commutative property of multiplication.

$\frac{b^{2}}{4 a} - b \frac{b}{2 a} + c$

Step 4.1.2.1.8

Multiply $- b \frac{b}{2 a}$ .

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Step 4.1.2.1.8.1

Combine $\frac{b}{2 a}$ and $b$ .

$\frac{b^{2}}{4 a} - \frac{b \cdot b}{2 a} + c$

Step 4.1.2.1.8.2

Raise $b$ to the power of $1$ .

$\frac{b^{2}}{4 a} - \frac{b^{1} b}{2 a} + c$

Step 4.1.2.1.8.3

Raise $b$ to the power of $1$ .

$\frac{b^{2}}{4 a} - \frac{b^{1} b^{1}}{2 a} + c$

Step 4.1.2.1.8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{b^{2}}{4 a} - \frac{b^{1 + 1}}{2 a} + c$

Step 4.1.2.1.8.5

Add $1$ and $1$ .

$\frac{b^{2}}{4 a} - \frac{b^{2}}{2 a} + c$

Step 4.1.2.2

To write $- \frac{b^{2}}{2 a}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{b^{2}}{4 a} - \frac{b^{2}}{2 a} \cdot \frac{2}{2} + c$

Step 4.1.2.3

Write each expression with a common denominator of $4 a$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.3.1

Multiply $\frac{b^{2}}{2 a}$ by $\frac{2}{2}$ .

$\frac{b^{2}}{4 a} - \frac{b^{2} \cdot 2}{2 a \cdot 2} + c$

Step 4.1.2.3.2

Multiply $2$ by $2$ .

$\frac{b^{2}}{4 a} - \frac{b^{2} \cdot 2}{4 a} + c$

Step 4.1.2.4

Combine the numerators over the common denominator.

$\frac{b^{2} - b^{2} \cdot 2}{4 a} + c$

Step 4.1.2.5

Simplify each term.

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Step 4.1.2.5.1

Simplify the numerator.

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Step 4.1.2.5.1.1

Factor $b^{2}$ out of $b^{2} - b^{2} \cdot 2$ .

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Step 4.1.2.5.1.1.1

Multiply by $1$ .

$\frac{b^{2} \cdot 1 - b^{2} \cdot 2}{4 a} + c$

Step 4.1.2.5.1.1.2

Factor $b^{2}$ out of $- b^{2} \cdot 2$ .

$\frac{b^{2} \cdot 1 + b^{2} (- 1 \cdot 2)}{4 a} + c$

Step 4.1.2.5.1.1.3

Factor $b^{2}$ out of $b^{2} \cdot 1 + b^{2} (- 1 \cdot 2)$ .

$\frac{b^{2} (1 - 1 \cdot 2)}{4 a} + c$

Step 4.1.2.5.1.2

Multiply $- 1$ by $2$ .

$\frac{b^{2} (1 - 2)}{4 a} + c$

Step 4.1.2.5.1.3

Subtract $2$ from $1$ .

$\frac{b^{2} \cdot - 1}{4 a} + c$

Step 4.1.2.5.2

Move $- 1$ to the left of $b^{2}$ .

$\frac{- 1 \cdot b^{2}}{4 a} + c$

Step 4.1.2.5.3

Move the negative in front of the fraction.

$- \frac{b^{2}}{4 a} + c$

Step 4.2

List all of the points.

$(- \frac{b}{2 a}, - \frac{b^{2}}{4 a} + c)$

Step 5