The ratio of the greatest value of 2 - cos x + sin2 x to its least value is ?

Question

The ratio of the greatest value of 2 - cos x + sin2 x to its least value is

1. 1/4

2. 9/4

3. 13/4

4. None of these

Answer 1

i learnd this in math last year  dang it lol im just gona guss with number 3

Answer 2

Assuming you mean 2 - cos x + sin²x, it's the third answer, 13/4

Differentiating gives

sin x + 2 sin x cos x

= sin x(1 + 2 cos x)

= 0 when sin x = 0 or cos x = -1/2

Hence x = 0, π, 2π/3, 4π/3

Calculating the function values at these points (no need to investigate which points are max, which are min:  The function is continuous and has no asymptotes, so the stationary points give all the possibilities for max or min) we get

1, 13/4, 3, 13/4

Hence the greatest value is 13/4, the least is 1, and so the ratio is

13/4.