Find the Local Maxima and Minima y=tan(x)

$y = \tan (x)$

Write $y = \tan (x)$ as a function.

$f (x) = \tan (x)$

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\sec^{2} (x)$

Find the second derivative of the function.

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

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To apply the Chain Rule, set $u$ as $\sec (x)$ .

$f'' (x) = \frac{d}{d u} (u^{2}) \frac{d}{d x} (\sec (x))$

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

$f'' (x) = 2 u \frac{d}{d x} (\sec (x))$

Replace all occurrences of $u$ with $\sec (x)$ .

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

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$f'' (x) = 2 \sec (x) (\sec (x) \tan (x))$

Raise $\sec (x)$ to the power of $1$ .

$f'' (x) = 2 (\sec (x) \sec (x)) \tan (x)$

Raise $\sec (x)$ to the power of $1$ .

$f'' (x) = 2 (\sec (x) \sec (x)) \tan (x)$

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

$f'' (x) = 2 {\sec (x)}^{1 + 1} \tan (x)$

Add $1$ and $1$ .

$f'' (x) = 2 \sec^{2} (x) \tan (x)$

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

$\sec^{2} (x) = 0$

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

$\sec (x) = \pm \sqrt{0}$

The complete solution is the result of both the positive and negative portions of the solution.

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Simplify the right side of the equation.

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Rewrite $0$ as $0^{2}$ .

$\sec (x) = \pm \sqrt{0^{2}}$

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

$\sec (x) = \pm 0$

$\pm 0$ is equal to $0$ .

$\sec (x) = 0$

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

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

No Local Extrema

Find the Local Maxima and Minima y=tan(x)

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