Help!!! Finding max, min, and inflection points given derivative

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kaitk

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The derivative of a function is f'(x)= (x-1)^2(x+3). Find the value of x at each point where f has a
a) local maximum
b) local minimum
c) point of inflection
Okay, so I know that I need to analyze the behavior of x at various intervals....
interval: x<1 sign y': + behavior: increasing
interval: -3<x<1 sign y': - behavior: decreasing
interval: -3<x sign y': + behavior: increasing
Is that right so far? If so, then I need to find the second derivative so that I can find when the graph is concave up/down. I'm not sure how to find the second derivative (product rule, chain rule?) It's kind of tricky. After I have the second derivative and the info on concave up/down, I can sketch a possible graph for f(x), right? Then I think I can answer the question. Can someone help me please?
 
kaitk said:
The derivative of a function is f'(x)= (x-1)^2(x+3). Find the value of x at each point where f has a
a) local maximum
b) local minimum
c) point of inflection
Okay, so I know that I need to analyze the behavior of x at various intervals....
interval: x<1 sign y': + behavior: increasing
interval: -3<x<1 sign y': - behavior: decreasing
interval: -3<x sign y': + behavior: increasing
Is that right so far? If so, then I need to find the second derivative so that I can find when the graph is concave up/down. I'm not sure how to find the second derivative (product rule, chain rule?) It's kind of tricky. After I have the second derivative and the info on concave up/down, I can sketch a possible graph for f(x), right? Then I think I can answer the question. Can someone help me please?
Click to expand...

Note that

x < -3 → we have f'(x) < 0 → f(x) is decreasing

x > -3 → we have f'(x) >0 → f(x) is increasing (except at x= 1, f'(x) = 0, but at x= 1+ or at x= 1- the function f'(x) is positive and f(x) is increasing)

But that really does not matter for this problem

Since you are looking for behavior of f(x) → find the roots of f'(x).

Next find the expression f"(x) find roots of that function.

f"(x) can be found by differentiating f'(x) [ = (x-1)^2(x+3)]

You will need to use product rule and chain rule of differentiation here:

you have

g(x) = [w(x)]^n * q(x)

g'(x) = n * [w(x)]^(n-1) * w'(x) * q(x) + [w(x)]^n * q'(x)

Please show us how far you can progress with this information.
 
So like this?
2(x-1) (1)(x+3) + (x-1)^2 (1)
I'm really confused. (I'm trying to teach all of this to myself, just using my book...)
 
kaitk said:
The derivative of a function is f'(x)= (x-1)^2(x+3). Find the value of x at each point where f has a
a) local maximum
b) local minimum
c) point of inflection
Okay, so I know that I need to analyze the behavior of x at various intervals....
interval: x<1 sign y': + behavior: increasing
interval: -3<x<1 sign y': - behavior: decreasing
interval: -3<x sign y': + behavior: increasing
Is that right so far? If so, then I need to find the second derivative so that I can find when the graph is concave up/down. I'm not sure how to find the second derivative (product rule, chain rule?) It's kind of tricky. After I have the second derivative and the info on concave up/down, I can sketch a possible graph for f(x), right? Then I think I can answer the question. Can someone help me please?
Click to expand...
This is a kind of backwards problem. If you answer the following questions without getting rattled, you will get your answer. Let us know where you get stuck if you do.

We have the equation for f'(x), from which f(x) which can be determined using integral calculus, but you may not have studied integral calculus yet.

Whether or not, however, you have studied integral calculus makes no difference because the question never asks you about any value of
f(x). It asks you at what values of x does f(x) have a local maximum, a local minimum, and a point of inflection.

Is f(x) differentiable?

If so, what value does its derivative have at a local maximum? At a local minimum? At a point of inflection?

Do you see a way to determine at what values of x the derivative of f(x) has that value?

How do you distinguish between those three possibilities?
 
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