there are no examples of this type of problem in my book so if you could help walk me through it - that would be extremely helpful. thanks ahead of time.
Find the extreme values of the function on the interval and where they occur.
4) F(x)=³√(x); -3</=x</=64
A. Maximum at (64, 4), and minimum at (-3, ³√-3)
B. Maximum at (-64, 4), and minimum at (0,0)
C. Maximum at (0,0), and minimum at (64,4)
D. Maximum at (64,4), and minimum at (-64,-4).
Thanks again.
There must be problems on extrema. The extrema occur where f' = 0. Here,
f(x) = x^(1/3)
f'(x) = 1/3 x^(-2/3)
f'(x) is never zero anywhere, especially in this interval.
So, we just want the maximum and minimum values attained on the interval. Note that f' > 0 everywhere, so it is strictly increasing. That means
f(-3) < f(64)
f(-3) = -∛3
f(64) = 4
So, those are the extrema on this interval.
Looks like (A) to me.
Thanks! I'll go with that and then if I do get it wrong - I'll ask for an explanation as to why from my teacher. Thanks for explaining it too :) It's starting to make more sense.
To find the extreme values of the function F(x)=³√(x) on the interval -3≤x≤64, we need to find the local maximum and minimum points.
To do this, we can start by finding the critical points of the function. Critical points occur where the derivative of the function is either zero or undefined.
First, let's find the derivative of F(x). Using the power rule, we have:
F'(x) = (1/3) * d/dx (x^(1/3))
= (1/3) * (1/3) * x^(-2/3)
= 1/(3x^(2/3))
Next, let's find the critical points by setting F'(x) equal to zero and solving for x:
1/(3x^(2/3)) = 0
This equation has no real solutions because the denominator can never be zero.
However, we also need to check the endpoints of the given interval, -3 and 64, since they could also potentially be extreme points.
Now, let's substitute the values of the endpoints into the original function F(x)=³√(x):
F(-3) = ³√(-3) = -√3
F(64) = ³√(64) = 4
So, the function has a minimum value of -√3 at x = -3 and a maximum value of 4 at x = 64.
Therefore, the correct answer choice is A. Maximum at (64, 4), and minimum at (-3, ³√-3).