Question

For the function f(x) = 20x3 − 3x5, determine the interval(s) for which f(x) increases.
Could someone also explain when interval increases or decreases

Answer 1

Differentiate the function with respect to x, and set the derivative to be greater than zero on some interval where the function is increasing.

Answer 2

so would it be 60x^2-15x^4>0 ?

Answer 3

yes

Answer 4

what would i do afterword?

Answer 5

factor out \(15x^2\) first to make your life simpler

Answer 6

then ignore that term completely, because \(15x^2>0\) for any \(x\) (except 0)

Answer 7

Your job here is to find the interval or intervals on which the function f(x) is INCREASING.

Use calculus!
1. Find the 1st derivative of f(x).
2. Set your result = to 0. Solve for x. There may be more than one x value that satisfies this equation. These are called CRITICAL VALUES. Finding Crit. Vals. is an essential skill.
3. plot these x values on a number line.
4. Determine (write out) all the intervals created by your two critical values.
Example: (-infinity, 5)
5. Choose a "test number" from within each of your intervals.
6. Subst. each such test number into the derivative f '(x).
Rules: If the deriv. is neg, f(x) is decreasing on that interval
If the deriv. is pos, f(x) is increasing (this is what we want)

7. List the intervals on which f(x) is increasing.
DONE