find the Local minimum or Maximum Of the Function

**ZOOZ** · December 11th 2010, 07:59 AM

how i find the Local minimum or Maximum?

(there is an option that there is no Local minimum or Maximum point in this Function)

BUT i need to show the way.

thanks.

**Soroban** · December 11th 2010, 08:41 AM

Hello, ZOOZ!

$\text{Find the local minimum or maximum (if any): }\;f(x) \:=\:\cot x - \tan x$

I did some simplifying first . . .

$\displaystyle f(x) \;=\; \cot x - \tan x \;=\;\frac{\cos x}{\sin x} - \frac{\sin x}{\cos x} \;=\;\frac{\cos^2\!x - \sin^2\!x}{\sin x\cos x}$

. . . . $\displaystyle =\;\frac{\cos2x}{\frac{1}{2}\sin2x} \;=\;2\cot2x$

Set the derivative equal to zero: . $f'(x) \;=\;-4\csc^2\!2x \;=\;0$

. . But $|\csc2x| \:\ge\:1$ . . . $f'(x)$ can never equal zero.

Therefore, $\,f(x)$ has no maximum or minimum.

**ZOOZ** · December 12th 2010, 12:11 AM

i think you mistake with Absolute minimum and Absolute maximum .

BUT to find Local minimum or Maximum we not must that the derivative will equal to zero.

but still thanks about your answer.

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