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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.

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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).

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  • $\begingroup$ 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? $\endgroup$ – The Statistician Jan 5 at 17:05
  • $\begingroup$ @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. $\endgroup$ – Joel Pereira Jan 6 at 15:58
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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

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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.)

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