- It was shown earlier that the n
^{th}root of a positive number exists whenever n is a positive integer. Let y = x^{1∕n}. Prove y^{′}=x^{(1∕n) −1}. - Now for positive x and p,q positive integers, y = x
^{p∕q}is defined by y =. Find and prove a formula for dy∕dx. - For 1 ≥ x ≥ 0, and p ≥ 1, show that
^{p}≥ 1 − px. Hint: This can be done using the mean value theorem. Define f≡^{p}− 1 + px and show that f= 0 while f^{′}≥ 0 for all x ∈. - Using the result of Problem 3 establish Raabe’s Test, an interesting variation on the
ratio test. This test says the following. Suppose there exists a constant, C and a number
p such that
for all k large enough. Then if p > 1, it follows that ∑

_{k=1}^{∞}a_{k}converges absolutely. Hint: Let b_{k}≡ k − 1 + C and note that for all k large enough, b_{k}> 1. Now conclude that there exists an integer, k_{0}such that b_{k0}> 1 and for all k ≥ k_{0}the given inequality above holds. Use Problem 3 to conclude thatshowing that

b_{k}^{p}is decreasing for k ≥ k_{0}. Thus≤ M∕b_{k}^{p}= M∕^{p}. Now use comparison theorems and the p series to obtain the conclusion of the theorem. - The graph of a function y = fis said to be concave up or more simply “convex” if wheneverandare two points such that y
_{i}≥ f, it follows that for each point,on the straight line segment joiningand,y ≥ f. Show that if f is twice differentiable on an open interval,and f^{′′}> 0, then the graph of f is convex. - Show that if the graph of a function f defined on an interval is convex, then if f
^{′}exists on, it must be the case that f^{′}is a non decreasing function. Note you do not know the second derivative exists. - Convex functions defined in Problem 5 have a very interesting property. Suppose
_{i=1}^{n}are all nonnegative, sum to 1, and suppose ϕ is a convex function defined on ℝ. ThenVerify this interesting inequality.

- If ϕ is a convex function defined on ℝ, show that ϕ must be continuous at every point.
- Prove the second derivative test. If f
^{′}= 0 at x ∈, an interval on which f is defined and both f^{′},f^{′′}exist and are continuous on this interval, then if f^{′′}> 0, it follows f has a local minimum at x and if f^{′′}< 0, then f has a local maximum at x. Show that if f^{′′}= 0 no conclusion about the nature of the critical point can be drawn. It might be a local minimum, local maximum or neither. - Recall the Bernstein polynomials which were used to prove the Weierstrass
approximation theorem. For f a continuous function on ,
It was shown these converge uniformly to f on

. Now suppose f^{′}exists and is continuous on. Show p_{n}^{′}converges uniformly to f^{′}on. Hint: Differentiate the above formula and massage to finally getThen form the

Bernstein polynomial for f^{′}and show the two are uniformly close. You will need to estimate an expression of the formwhich will be easy to do because of the mean value theorem and uniform continuity of f

^{′}. - In contrast to Problem 10, consider the sequence of functions
Show it converges uniformly to f

≡ 0. However, f_{n}^{′}converges to 1, not f^{′}. Hint: To show the first part, find the value of x which maximizes the function. You know how to do this. Then plug it in and you will have an estimate sufficient to verify uniform convergence.

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