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 ∈
The graph of a function y = f
is said to be concave up or more
simply “convex” if whenever
are two points such that
, it follows that for each point,
on the straight line segment
,y ≥ f
. Show that if f is twice differentiable
on an open interval,
> 0, then the graph of f is convex.
Suppose you have a function f which has two derivatives. Suppose f′′> 0.
Give a sketch of the graph of f. In particular, show that the overall shape of
the function is that it curves up. See the next problem.
Show that if the graph of a function f defined on an interval
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 2 have a very interesting property. Suppose
i=1n are all nonnegative, sum to 1, and suppose ϕ is a convex function defined
on ℝ. Then
(∑n ) n∑
ϕ akxk ≤ akϕ(xk).
Verify this interesting inequality.
Find all critical points of the function
f (x) = 1x4 − 2x3 + 11x2 − 6x
Classify each critical point according to whether it is a local minimum, maximum,
. Describe the critical points and classify
≡ x4− 6x3 + 12x2− 10x + 3. Find and classify the critical points.