Calculus Examples

$f (x) = - x^{2} + 2 x + 6$

Find the inflection points.

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Find the second derivative.

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Find the first derivative.

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

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

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

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Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

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

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

Multiply $2$ by $- 1$ .

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

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

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

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

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

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

Multiply $2$ by $1$ .

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

Differentiate using the Constant Rule.

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Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$f' (x) = - 2 x + 2 + 0$

Add $- 2 x + 2$ and $0$ .

$f' (x) = - 2 x + 2$

Find the second derivative.

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

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

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

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

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

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

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

Multiply $- 2$ by $1$ .

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

Differentiate using the Constant Rule.

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

$f'' (x) = - 2 + 0$

Add $- 2$ and $0$ .

$f'' (x) = - 2$

The second derivative of $f (x)$ with respect to $x$ is $- 2$ .

$- 2$

Since $- 2 \neq 0$ , there are no solutions.

No solution

No values found that can make the second derivative equal to $0$ .

No Inflection Points

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:

${x | x \in ℝ}$

The graph is concave down because the second derivative is negative.

The graph is concave down

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