What are the critical points of # f(x) = x*sqrt(8-x^2) #?
To find the critical points of a function, find where the first derivative equals 0.
You need to use the product rule for derivatives:
where in this case,
It is easier to represent
Now use the Chain Rule to find the derivative of
(Don't forget to find the derivative of the inside function --
that's where the
Clean this up so
OKAY, put it all together now:
Combine these two fractions into one by getting common denominators and you get
Critical values occur when
numerator = 0 and/or denominator = 0).
Setting the numerator = 0 and solving results in
Setting the denominator = 0 and solving results in
If you use Symbolab, you can quickly get these answers, I hope my explanation helped! :)