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