# Calculus

#4

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?

Yes

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

#7

f(x) = sin2x - cos x, and 0 <= x <= π

#9

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?

https://brainmass.com/math/calculus-and-analysis/happens-concavity-functions-added-190574

#### Solution Preview

#4

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?

When functions are added, the concavity of the sum ...

#### Solution Summary

The expert determines what happens to concavity when functions are added. Neat, step-by-step solutions using graphs are provided.