differentiation

Question

The function f is defines by f(x)=sinx + cosx , for 0≤x≤2π.

Find (i) The range of values of x for which f(x) is increasing

       (ii) The range of values of x for which f(x) is decreasing.

 

Ans: (i) 0<x<π or 5π/4 <x< 2π

        (ii) π/4 <x<5π/4

 

Thank you!

Answer 1

Find when the derivative is equal to 0. This will break up the interval 0 to 2pi into sections that are positive or negative. if the first derivative is positive, the function is incr4asing. If the first derivative is negative, the function is decreasing. so f(x) = sin x + cos x f'(x) = cos x - sin x cos x - sin x = 0 cos x = sin x x = 45 and x = 225 or pi/4 and 5pi/4 so you have the intervals 0 to pi/4, pi/4 to 5pi/4 snf 5/4 to 2pi. plug in a calue in each inerval to find out if the entire interval is positive or negative. f'(pi/6) = sqrt3 / 2 - 1/2 positive so f(x) is increasing f(pi) = -1 - 0 negative so f(x) is decreasing f(3pi/2) = 0 - (-1) positive so f(x) is increasing Hope this helped.