Applications of Differentiation

Question

Consider the function f(x)= 2sin(x) + [(sqrt3)/2] x² This function has two inflection numbers A<B in [0,2pi]:

A= and B= For each of the following intervals, tell whether f(x) is concave up (type in CU) or concave down (type in CD).

[0,A)=

(A,B)=

(B,2pi]=

Answer 1

The graph of f(x) will be concave up on any interval for which f^"(x) is + and concave down when f^"(x) is - . Thus, the graph of f will have a point of inflection any time f^" changes signs:

f^'(x) = 2cosx + √3·x

f^"(x) = -2sinx +√3 = 0

sinx = √3 / 2 ; x = 2π / 3 or 4π / 3.

A = 2π / 3 , B = 4π / 3 ; f^"(x) is + on (0 , 2π / 3) and (4π / 3 , 2π). It is - on (2π / 3 , 4π / 3).

Thus, f is CU on (0 , A) and (B , 2π) while f is CD on (2π / 3 , 4π / 3).