# What are the critical points of #f(x)=x^(2/3) *e^ (x^2-4)#?

##### 1 Answer

#### Answer:

there is only one critical point, at x = 0

#### Explanation:

critical points are the points in a function where the slope changes from positive to negative or vice versa, therefore these points have potential to be the maximum or minimum point of a function.

this change of slop can happen at two types of situations, where the derivative is 0 and where it is undefined these places can also be the maximum or minimum value.

to find the critical points, we have to see where the derivative is 0 or undefined so you need to differentiate the function and

the derivative is

you can see that it will be undefined at

so 1 critical point is

to get the derivative equal to zero the two terms (

after eliminating the same terms which are

we are left with

multiplying both sides by

now Unless we are Using complex numbers

a square of any number cannot be negative therefore there doesn't exist any real value of x which makes the derivative equal to 0, only undefined at x = 0

therefore the only critical point is where x = 0

you can see this on the graph of the function graph{x^(2/3) * e^(x^2 - 4) [-11.25, 11.25, -5.625, 5.625]}

the only place where the derivative or the slope changes from negative to positive s at x = 0