find the Local minimum or Maximum Of the Function

• Dec 11th 2010, 07:59 AM
ZOOZ
find the Local minimum or Maximum Of the Function
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.
• Dec 11th 2010, 08:41 AM
Soroban
Hello, ZOOZ!

Quote:

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

I did some simplifying first . . .

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

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

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

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

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

• Dec 12th 2010, 12:11 AM
ZOOZ
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.