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.
Best Answer
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).