Question

Show that the function f(x) = -x + cos x is decreasing.​

Answer 1

For a function to be decreasing, f'(x) <0

f(x) = -x + cos(x)

=> f'(x) = (-x)' + {cos(x)}'

=> f'(x) = -1 - sin(x)

=> f'(x) = -(1+sin(x))

.

Now,

-1 ≤ sin(x) ≤ 1

add 1 to the above inequality,

0 ≤ 1 + sin(x) ≤ 2

multiply the above inequality by -1,

0 ≥ -(1+sin(x)) ≥ -2

0 ≥ f'(x) ≥ -2

.

=> f'(x) lies between [-2,0]

=> f'(x) ≤ 0

Thus, f(x) is a decreasing function.

Therefore, -x + cos(x) is a decreasing function.