Inverse of functions

Question

Given that f(−7) = −3 and f'(x) ≤ 2, what is the largest possible value of f(0) if f(x) is given as a simple polynomial?

(b) Show that f(x) = x^3 −7x^2 +25x+8 has exactly one real root.

(c) Determine whether or not the MVT can be applied to the following function on the closed interval [0,π]. If so, find all possible values of c. f(x) = sin2x.

(d) Use the MVT to show that |sin w −sin z| ≤ |w −z| for 0 ≤z <w≤2π.

(e) Convince yourself that f(x) satisfies the Mean Value Theorem on ONLY one interval below: f(x) =x^2, if x < 2

6 −x, if x≥2

a [−∞,1)

b [0,4]

c [0,2)

d [2,∞)

Answer 1

f(x) = x^3 -7x^2 +25x + 8

f(0) =8

f(-1) =-25

there one negative real solution between x=0 and x=-1. it's about x=-0.3

by Descartes Rule of Signs, there is a maximum one negative real solution

and a maximum 2 positive real solutions

the real solutions are either 2 or 0

the other two solutions are imaginary

use a graphing calculator. f(x) only intersects the x axis once.

or use an online calculator for solving cubic equations,

it gives one real and 2 imaginary solutions.