Increasing and Decreasing functions using interval notation

I have a question where I am asked to indicate in "interval notation" when a given function is increasing or decreasing.

I just don't understand what I'm meant to be doing, as I have been given an interval, plus if I try to find the x-values when I put my function $$f'(x)>0$$ and $$f'(x)<0$$, I always get no x values out as the x-values are all in the real numbers.

This is what I have been given...

For the function: $$g(x)=e^t$$where $$t=sin(x)$$.

On the interval $$[0, 4\pi]$$, indicate in interval notation when it is increasing and when it is decreasing.

How am I meant to do this question? Any help would be most appreciated.

Thanks.

So this is a question about the sign of the derivative. Recall that if $$f^{\,\prime} >$$ 0, then f is increasing whereas if $$f^{\prime}$$ $$<$$ 0, then f is decreasing. So the first step is to find f$$^{\,\prime}$$:

$$f = e^{\sin(x)} \text{ on [0,4\pi]}$$ $$f^\prime = \cos(x)e^{\sin(x)}$$

Now you first want to find the critical points where $$f^\prime$$ = 0. In this case, this only occus when $$\cos(x)$$ = 0 in [0,4$$\pi$$], namely $$\left\{\frac{\pi}{2},\frac{3\pi}{2},\frac{5\pi}{2},\frac{7\pi}{2}\right\}$$.

Now you break up the interval using the critical points as endpoints of your partition. Then you take sample values from each partition and plug them into $$f^\prime$$. The sign of the derivative will be the same for any value in a given partition.In this case, we need only check the sign of $$\cos(x)$$ since $$e^{\sin(x)}$$ is never 0.

The subintervals where $$f^\prime$$ > 0 (resp. < 0 ) is where f is increasing (resp. decreasing).

• So with the critical values, I substitute each of the 4 critical points into $f'(x)$ and this will give me which points are increasing/decreasing on the given interval. Have I understood this or got this completely wrong? – The Statistician Jan 5 at 17:05
• @TheStatistician Yes. when you pick a sample point $x^\star$ in some subinterval S, and find the sign of $f^{\,\prime}(x^{\star})$, then every point in S has the same sign. – Joel Pereira Jan 6 at 15:58

Calculate the derivative of g, $$g'(x)=cos(x)e^{sin(x)}$$, now discuss the sign of g' , if g'(x) is positive then g is increasing, if g' is negative then g is decreasing

Hint: Using the Chain Rule, you get

$$f’(x) = \cos(x)e^{\sin(x)}$$

Clearly $$e^{\sin(x)} > 0$$ for all $$x$$, so the real question is about where $$\cos(x) > 0$$ and $$\cos(x) < 0$$. (Recall the unit circle and the quadrants.)