Kenneth Kuttler

152 CHAPTER 7. THE DERIVATIVE

showing |ak|bpk is decreasing for k≥ k0. Thus |ak| ≤M/bp

k = M/(k−1+C)p . Nowuse comparison theorems and the p series to obtain the conclusion of the theorem.

5. The graph of a function y = f (x) is said to be concave up or more simply “convex” ifwhenever (x1,y1) and (x2,y2) are two points such that yi ≥ f (xi) , it follows that foreach point, (x,y) on the straight line segment joining (x1,y1) and (x2,y2) ,y≥ f (x) .Show that if f is twice differentiable on an open interval, (a,b) and f ′′ (x)> 0, thenthe graph of f is convex.

6. Show that if the graph of a function f defined on an interval (a,b) is convex, then iff ′ exists on (a,b) , it must be the case that f ′ is a non decreasing function. Note youdo not know the second derivative exists.

7. Convex functions defined in Problem 5 have a very interesting property. Suppose{ai}n

i=1 are all nonnegative, sum to 1, and suppose φ is a convex function defined onR. Then

∑k=1

akxk

)≤

∑k=1

akφ (xk) .

Verify this interesting inequality.

8. If φ is a convex function defined on R, show that φ must be continuous at everypoint.

9. Prove the second derivative test. If f ′ (x) = 0 at x ∈ (a,b) , an interval on which f isdefined and both f ′, f ′′ exist and are continuous on this interval, then if f ′′ (x)> 0, itfollows f has a local minimum at x and if f ′′ (x)< 0, then f has a local maximum atx. Show that if f ′′ (x) = 0 no conclusion about the nature of the critical point can bedrawn. It might be a local minimum, local maximum or neither.

10. Recall the Bernstein polynomials which were used to prove the Weierstrass approxi-mation theorem. For f a continuous function on [0,1] ,

pn (x) =n

∑k=0

(nk

)f(

)xk (1− x)n−k

It was shown these converge uniformly to f on [0,1] . Now suppose f ′ exists and iscontinuous on [0,1] . Show p′n converges uniformly to f ′ on [0,1] . Hint: Differentiatethe above formula and massage to finally get

p′n (x) =n−1

∑k=0

(n−1

)(f( k+1

)− f

( kn

)1/n

)xk (1− x)n−1−k .

Then form the (n−1) Bernstein polynomial for f ′ and show the two are uniformlyclose. You will need to estimate an expression of the form

f ′(

kn−1

)−

f( k+1

)− f

( kn

)1/n

which will be easy to do because of the mean value theorem and uniform continuityof f ′.

15210.CHAPTER 7. THE DERIVATIVEshowing |a,|b? is decreasing for k > ko. Thus |az| << M/b? =M/(k—1+C)?. Nowuse comparison theorems and the p series to obtain the conclusion of the theorem.. The graph of a function y = f (x) is said to be concave up or more simply “convex” ifwhenever (x,y) and (x2, y2) are two points such that y; > f (x;) , it follows that foreach point, (x,y) on the straight line segment joining (x1, y1) and (x2,y2),y > f(x).Show that if f is twice differentiable on an open interval, (a,b) and f” (x) > 0, thenthe graph of f is convex.Show that if the graph of a function f defined on an interval (a,b) is convex, then iff’ exists on (a,b) , it must be the case that f’ is a non decreasing function. Note youdo not know the second derivative exists.Convex functions defined in Problem 5 have a very interesting property. Suppose{a;};_, are all nonnegative, sum to 1, and suppose @ is a convex function defined onR. Then@ e a] < PV ard (x).k=l k=1Verify this interesting inequality.If @ is a convex function defined on R, show that @ must be continuous at everypoint.Prove the second derivative test. If f’ (x) = 0 at x € (a,b), an interval on which f isdefined and both f’, f” exist and are continuous on this interval, then if f” (x) > 0, itfollows f has a local minimum at x and if f” (x) <0, then f has a local maximum atx. Show that if f” (x) = 0 no conclusion about the nature of the critical point can bedrawn. It might be a local minimum, local maximum or neither.Recall the Bernstein polynomials which were used to prove the Weierstrass approxi-mation theorem. For f a continuous function on [0,1],mid=¥ (te MananIt was shown these converge uniformly to f on [0,1]. Now suppose /” exists and iscontinuous on [0, 1]. Show p’, converges uniformly to f’ on [0,1]. Hint: Differentiatethe above formula and massage to finally getn—-1 n— k+l) kPh, (x) = y ( k ') (io) (1 ax) kThen form the (n— 1) Bernstein polynomial for f’ and show the two are uniformlyclose. You will need to estimate an expression of the form(7)which will be easy to do because of the mean value theorem and uniform continuityof f’.