Find the intervals of increase and decrease for the following function
h(x)=(x^3)(x-1)^4
h'(x)=[(3x^2)(x-1)^4]+(4x^3)(x-1)^3
How do I find the zeros?
Thanks.
The zeros of the derivative or the zeros of the function ?
I will assume you want the zeros of the derivative.
factor it first
= x^2(x-1)^3[4x + 3(x-1)]
= x^2(x-1)^3(7x-3)
we set this equal to zero to get
x = 0, x = 1 and x = 3/7
remember that the function increases when the first derivative is positive, and decreases when that derivative is negative.
Can you take it from here ?
Do you factor it by taking the lowest ^ ? I can continue from there, thanks! :)
The answer key indicated that 0 is not a zero.
Your question was,
"how do I find the zeros"
x = 0 is a solution to h(x) = 0 and h'(x) = 0
If x=0, shouldn't it be used to determine the increase and decrease?
To find the intervals of increase and decrease for the given function, you need to analyze the signs of its derivative, h'(x). Intervals where h'(x) is positive represent intervals of increase, while intervals where h'(x) is negative represent intervals of decrease.
To find the zeros of h'(x), which correspond to critical points or possible inflection points of the function h(x), we need to solve the equation h'(x) = 0.
So, let's solve for h'(x) = 0:
[(3x^2)(x-1)^4]+(4x^3)(x-1)^3 = 0
First, we can factor out a common term of (x-1)^3 from each term:
(x-1)^3 [(3x^2)(x-1) + 4x^3] = 0
Now, we have (x-1)^3 as a common factor. So, either (x-1)^3 = 0 or [(3x^2)(x-1) + 4x^3] = 0.
1. Solving (x-1)^3 = 0:
(x-1)^3 = 0 implies x-1 = 0, since any number cubed and equal to 0 must be 0.
So, x - 1 = 0 gives us x = 1.
2. Solving [(3x^2)(x-1) + 4x^3] = 0:
To solve this equation, we distribute and rearrange the terms:
3x^3 - 3x^2 + 4x^3 = 0
7x^3 - 3x^2 = 0
Factoring out a common term of x^2:
x^2(7x - 3) = 0
This equation is satisfied if either x^2 = 0 or 7x - 3 = 0.
a) Solving x^2 = 0:
x^2 = 0 implies x = 0.
b) Solving 7x - 3 = 0:
7x - 3 = 0 implies 7x = 3, and dividing both sides by 7 gives x = 3/7.
So, the zeros of h'(x) are x = 1, x = 0, and x = 3/7.
Now, to determine the intervals of increase and decrease, we need to create a sign chart by examining intervals between these critical points and outside them.
Check the sign of h'(x) in the intervals (-∞, 0), (0, 3/7), (3/7, 1), (1, ∞). Plug any test point from each interval into h'(x) to determine the sign.
For example, taking a test point in the interval (-∞, 0), such as x = -1, evaluating h'(-1) will give you a positive value. Repeat this process for the remaining intervals to determine the signs of h'(x) in each interval.
Based on the signs of h'(x) in the intervals, you can determine which intervals represent intervals of increase and which represent intervals of decrease for the function h(x)=(x^3)(x-1)^4.