1.Let f be differentiable on an open interval I and let a,b in I with a< b. Suppose that m is between f'(a) and f'(b). Then there is a number c in (a,b) such that f'(c)=m. (Hint: consider g(x)=f(x)-mx.)
2. Suppose f:R->R is continuous. Let n be a positive real number, and assume that for every x in R and a>0, f(ax)=(a^n)f(x).
(a) If n > 1 show that f is differentiable at 0.
(b) If 0 < n < 1 show that f is not differentiable at 0.
(c) If n=1, show that f is differentiable at 0 if and only if it is linear.
(Hint: what is f(0)?)
3. Use the mean value theorem to show that .99^5> =.95
2. Suppose f:R->R is continuous. Let n be a positive real number, and assume that for every x in R and a>0, f(ax)=(a^n)f(x).
(a) If n > 1 show that f is differentiable at 0.
(b) If 0 < n < 1 show that f is not differentiable at 0.
(c) If n=1, show that f is differentiable at 0 if and only if it is linear.
(Hint: what is f(0)?)
3. Use the mean value theorem to show that .99^5> =.95
