Q1) True or false? If the statement is true, explain how. If the statement is fa

Question

Q1) True or false? If the statement is true, explain how. If the statement is false, give a counterexample.

     If f'' and g'' exist, and f and g are concave up for all x, then f(g(x)) is concave up for all x.

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Q2) Give an example of a function f(x) such that f''(x)=-f(x).

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Q3) Find  dz/dt  if     (z=(te^{3t}+e^{5t})^{9})     

Explanation / Answer

a) let me prove this with a very simple illustration.

See it has already been given that:

If f'' and g'' exist, and f and g are concave up for all x.

so let's assume f(x) is concave for all x.

and f(y) should also be concave for all x.

now let y=g(x)

therefore we have

f(y) = f(g(x)) concave for all x. This simply means that concavity or convexity depends on outer function and not inner function. Hope it's clear.

b) simple f(x) = sinx

we have

f'(x) = cosx

and f''(x) = -sinx = -f(x)

c) dz/dt = 9 (te^3t+e^5t)^8 * [e^3t + 3te^3t + 5e^5t]

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