Finding global maximum or minimum

Question

Assignment:

Q1. What happens to concavity when functions are added?

 

a) If f(x) and g(x) are concave up for all x, is f(x) + g(x) concave up for all x?

b) If f(x) is concave up for all x and g(x) is concave down for all x, what can you say about the concavity of f(x) + g(x)?  For example, what happens if f(x) and g(x) are both polynomials of degree 2?

c) If f(x) is concave up for all x and g(x) is concave down for all x, is it possible for f(x) + g(x) to change concavity infinitely often?

Find the value(s) of x for which:

a) f(x) has a local maximum or local minimum.  Indicate which ones are maxima and which are minima.

b) f(x) has a global maximum or global minimum

Q2. f(x) = sin²x – cos x, and 0 <= x <= π

Q3. A line goes through the origin and a point on the curve y = x²e^-3x, for x>=0.  Find the maximum slope of such a line.  At what x-value does it occur?

Provide complete and step by step solution for the question and show calculations and use formulas.