Josephood7

Consider the function f(x)=7−7x2/3f(x)=7−7x2/3 on the interval [−1,1][−1,1].

Which of the three hypotheses of Rolle's Theorem fails for this function on the inverval?

(a) f(x)f(x) is continuous on [−1,1][−1,1].
(b) f(x)f(x) is differentiable on (−1,1)(−1,1).
(c) f(−1)=f(1)f(−1)=f(1).

Suppose that f(0)=−5f(0)=−5 and f′(x)≤4f′(x)≤4 for all values of x.x. Use the Mean Value Theorem to determine how large f(4)f(4) can possibly be.

Answer: f(4)≤f(4)≤ 

Suppose that f(t)f(t) is continuous and twice-differentiable for t≥0t≥0. Further suppose f′′(t)≤9f″(t)≤9 for all t≥0t≥0 and f(0)=f′(0)=0f(0)=f′(0)=0.

Using the Racetrack Principle, what linear function g(t)g(t) can we prove is greater than f′(t)f′(t) (for t≥0t≥0)?
g(t)=g(t)= 

Then, also using the Racetrack Principle, what quadratic function h(t)h(t) can we prove is greater than than f(t)f(t) (for t≥0t≥0)?
h(t)=h(t)= 

For both parts of this problem, be sure you can clearly state how the theorem is applied to prove the indicated inequalities.

Solution:

Consider the functions f(x)=ex−1−1f(x)=ex−1−1 and g(x)=x−1g(x)=x−1. These are continuous and differentiable for x>0x>0. In this problem we use the Racetrack Principle to show that one of these functions is greater than the other.

(a) Find a point cc such that f(c)=g(c)f(c)=g(c). c=c= 

(b) Find the equation of the tangent line to f(x)=ex−1−1f(x)=ex−1−1 at x=cx=c for the value of cc that you found in (a).
y=y= 

(c) Based on your work in (a) and (b), what can you say about the derivatives of ff and gg?
f′(x)f′(x)  ? < <= = >= >  g′(x)g′(x) for 0<x<c0<x<c, and
f′(x)f′(x)  ? < <= = >= >  g′(x)g′(x) for c<x<∞c<x<∞.

(d) Therefore, the Racetrack Principle gives
f(x)f(x)  ? < <= = >= >  g(x)g(x) for x≤cx≤c, and
f(x)f(x)  ? < <= = >= >  g(x)g(x) for x≥cx≥c.