Question

Find the inflection points of the function ( f(x) = 8sin(x) + cot(x) ), ( -π ≤ x ≤ π ).

a) ( x = -π/2 ), ( x = π/2 )
b) ( x = -π ), ( x = 0 )
c) ( x = 0 ), ( x = π )
d) ( x = -π/2 ), ( x = 0 )

Answer 1

There are no inflection points in the given interval.

Step-by-step explanation:

To find the inflection points of the function f(x) = 8sin(x) + cot(x) on the interval -π ≤ x ≤ π, we need to find where the concavity changes. The concavity changes when the second derivative of the function equals zero or is undefined.

Taking the second derivative of f(x), we get:

f''(x) = -8sin(x) + csc^2(x)

Setting f''(x) equal to zero, we find:

-8sin(x) + csc^2(x) = 0

This equation does not have any solutions on the given interval. Therefore, there are no inflection points in the interval -π ≤ x ≤ π.