Consider the function f(x)=3cosx−cos^(3)x for 0<x<2π.
For the following questions, write inf for ∞, -inf for −∞, U for the union symbol, None if no answer exists, and separate by a comma if more than one answer exist.
a.) The x-intercepts are .
b.) f′(x)= .
c.) f(x) is increasing on the interval(s) .
d.) f(x) is decreasing on the interval(s) .
e.) f(x) has a local maximum at x = .
f.) f(x) has a local minimum at x = .
g.) f′′(x)= .
h.) f(x) is concave up on the interval(s) .
i.) f(x) is concave down on the interval(s) .
j.) The x-coordinate of the points of inflection are
a.) The x-intercepts are None.
b.) f′(x) = -3sin(x) + 3cos^2(x)sin(x)
c.) f(x) is increasing on the interval(s) (0, π/2)U(3π/2, 2π).
d.) f(x) is decreasing on the interval(s) (π/2, 3π/2).
e.) f(x) has a local maximum at x = π/2.
f.) f(x) has a local minimum at x = 3π/2.
g.) f′′(x) = -3cos(x) - 3sin(x)cos^2(x) + 6sin^2(x)cos(x)
h.) f(x) is concave up on the interval(s) (0, π/2)U(3π/2, 2π).
i.) f(x) is concave down on the interval(s) (π/2, 3π/2).
j.) The x-coordinate of the points of inflection are None.