First Derivative Test for f(x) = (1-x)^2(1+x)^3: Extrema, Intervals, and Values

In summary, the conversation discusses finding the relative extrema, values of f at those points, and intervals of increasing and decreasing for the function f(x) = (1-x)^2(1+x)^3. The correct differentiation is given and it is determined that the function has zeros at x=1 and x=-1, with another zero revealed after simplifying the expression. The need to apply the chain rule is also mentioned.
  • #1
jhodzzz
15
0

Homework Statement


I have to get the following:
- relative extrema of f
- values of f at which the relative extrema occurs
- intervals on which f is increasing
- intervals on which f is decreasing

when f(x) = (1-x)2 (1+x)3


Homework Equations


Now when get to have the first derivative by multiplication rule f'(x) = g(x)*h'(x)+h(x)*g'(x):
f'(x) = ((1-x)2)(3(1+x)2)+((1+x)3)(2(1-x))
is it correct to say that f'(x)=0 when x=1 or x=-1?

and if it is, by substituting 1 and -1 to f(x), i'll arrive on ordered pairs' (1,0),(-1,0) which are on a vertical line. when i checked if the interval -1 < x < 1 is increasing or decreasing, i arrived at an answer that it is increasing which is not possible considering the locations of the two critical points.

Where did I go wrong?
 
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  • #2
first, you need to differentiate correctly. the second differentiated term needs a what by composition...?
 
  • #3
you mean this differentiation: f'(x)=((1-x)2)(3(1+x)2)+((1+x)3)(2(1-x))

I arrived at that considering f(x)=g(x)*h(x) such that g(x)=(1-x)2 and h(x)=(1+x)3

so applying the multiplication rule, i should have that answer.
Do I still need to simplify it further?? will the factors vary by then?
 
  • #4
no...more like f(x)=g(h(x))*m(n(x))
 
  • #5
You need to apply the chain rule to get the second term correct.

You are correct that the function has zeros as x=1 and x=-1, but there's another zero that you won't see until you simplify the expression.
 

What is the First Derivative Test?

The First Derivative Test is a mathematical tool used to determine the nature of a critical point on a curve based on the sign of the derivative at that point.

How do you use the First Derivative Test?

To use the First Derivative Test, you first find the critical points of a function by setting the derivative equal to zero. Then, you evaluate the sign of the derivative at each critical point to determine whether it is a local maximum, local minimum, or neither.

What is the significance of the First Derivative Test?

The First Derivative Test is significant because it allows us to determine the behavior of a function at critical points without having to graph the function or use a table of values. This makes it a powerful tool for analyzing functions and understanding their behavior.

What are some common mistakes when using the First Derivative Test?

Some common mistakes when using the First Derivative Test include not finding all of the critical points, evaluating the derivative incorrectly, and not considering both sides of the critical point when determining the nature of the point.

Can the First Derivative Test be used to find global extrema?

No, the First Derivative Test can only be used to find local extrema. To find global extrema, you must also consider the behavior of the function at the endpoints of the interval.

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