# Thread: local max and min

1. ## local max and min

hi

have function $g(x)=cosxexp(\frac{2}{3}sinx)$

has derivative $g'(x)=(2-3sinx-2sin^2x)exp(\frac{2}{3}sinx)$

i factor out 1/3 and ignored $exp(\frac{2}{3}sinx)$ as it always positive and got $(1-2sinx)(2+sinx)$ then set it to zero and got x= $\frac{\pi}{6}$ and $\frac{5\pi}{6}$ used these to get coordinates, which i am happy with but when i try to find the max and min locals of g(x) i get the oposite result to the graph of g(x). no matter how i try to look at it i get $\frac{\pi}{6}$, $\frac{\sqrt{3}}{2}$ as minimum point and $\frac{5\pi}{6}$, $\frac{-\sqrt{3}}{2}$ as a maximum point. have been using first derivitive test as i am unable to work out how to get g''(x)

any help would be great

2. I dont know how you conducted the first devaritive test, but observe that g'(x)>0 for 0<x<(pi/4)
and g'(x)<0 for pi/4<x<5pi/4. Cos(x) decides the sign because exp(2/3sin(x)) is always > 0.

That means that on the interval 0<x<(pi/4) there is a local maxima, and on the interval pi/4<x<5pi/4, there is a local minima.

Looks like you just turned that one around.

3. Originally Posted by smartcar29
hi

have function $g(x)=cosxexp(\frac{2}{3}sinx)$

has derivative $g'(x)=(2-3sinx-2sin^2x)exp(\frac{2}{3}sinx)$

i factor out 1/3 and ignored $exp(\frac{2}{3}sinx)$ as it always positive and got $(1-2sinx)(2+sinx)$ then set it to zero and got x= $\frac{\pi}{6}$ and $\frac{5\pi}{6}$ used these to get coordinates, which i am happy with but when i try to find the max and min locals of g(x) i get the oposite result to the graph of g(x). no matter how i try to look at it i get $\frac{\pi}{6}$, $\frac{\sqrt{3}}{2}$ as minimum point and $\frac{5\pi}{6}$, $\frac{-\sqrt{3}}{2}$ as a maximum point. have been using first derivitive test as i am unable to work out how to get g''(x)

any help would be great
1. The roots of the equation

$2-3 \sin(x)-2\sin^2(x)=0$

corresponding to your first factor are $x=\pi/6 + 2 \pi n$ and $x=2 \pi/3+2 \pi m$, for all $n, \ m \in \mathbb{Z}$, and similarly there are multiple roots corresponding to the other factor.

2. How are you determining if a root corresponds to a maximum or minimum?

3. How are you evaluating $g(x)$?

CB