#2a f ' (x) >0 for all x hence increasing and one to one
Use the theorem on the derivative of the inverse:
if f(a) = c
Then if g = f^(-1)
g ' (c) = 1/ (f ' (a))
f(x) = 4x^3 + 5x + 1
f(0) = 1
g ' (1) = 1/ f'(0) = 1/ 5
Hi guys. Thanks in advance for the help last time, and I'm wondering if you guys could help me out here.
1. (Work problem)
The Drain-o-Max above ground pool pumping system is designed to do up to 50000(pi) ft-lbs of work on a single charge. It attaches directly to the side of your above ground pool and pumps the water right out over the side of the pool.
What is the radius of the largest 4 foot tall circular above ground pool that the Drain-o-Max can empty on a single charge? (Assume pool is initially full to the brim and the weight density of water is 62.5 ft-lbs.)
2. a) show that f(x) = 4x^3 + 5x + 1 is one-to-one.
b) Find g'(1), where g is the inverse function to f(x) = 4x^3 + 5x + 1.
For #1, I'm really lost and have no idea where to begin my answer.
For #2, for a), I found the derivative and then saw that it was a parabola. Is there something to do with the Mean Value Theorem to prove that the original eqn. is 1-to-1?
I have no idea how to do b either.
Thanks in advance guys!
VonNemo, I understand what you're doing.. but how would you get (pi)x^2 in terms of y? I clearly understand the 4-y part as the distance.
Perhaps my picture is just drawn completely wrong and I can't visually understand what is happening.