Originally Posted by

**nahduma** Actually, there are TWO critical values for $\displaystyle 0\leq x\leq 2\pi :\frac{\pi}{4} and \frac{9\pi}{4} $

Now, the function is increasing if the slope is positive and vice versa.

For all $\displaystyle x \in [0,\frac{\pi}{4}] and x \in [\frac{9\pi}{4}, 2\pi] , f'(x) \geq 0. similarly, f'(x) \leq 0 \forall x \in [\frac{\pi}{4},\frac{9\pi}{4}] $.

I hope you can solve the rest of the problem from this:

the local minima and maxima are situated at the critical points.

if the second derivative is positive, the graph is convex. if it is negative, it is concave.

the point of inflection is where the second derivative changes signs (ie. from +ve to -ve or vice versa). therefore, it becomes zero.