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