The function f is defined by the power series:

for all real numbers x.

(a) Find f'(0) and f''(0). Determine whether f has a local minimum, a local maximum, or neither at x = 0. Give a reason for your answer.

f(0) = 1
f'(0) = 0
f''(0) = -1/3

f has a local maximum at x = 0, because the first derivative is equal to zero and the second derivative is negative clearly indicating a local maximum.

(b) Show that 1 - ( 1 / 3! ) approximates f(1) with error less than ( 1 / 100 ).

Find error term by using the next term in the series, which is the term
( x^4 / 5! ), where x = 1.

(c) Show that y =f(x) is a solution to the differential equation xy' + y = cos x.