Question

Let f(x) = {– b² + (a – 1)b – 2} x + ∫(sin²x + cos⁴x)dx, a, b ∈ R If f(x) be an increasing function, find all the permissible values of a.

Answer 1

We have f(x) = {–b² + (a – 1) b – 2}x + ∫(sin²x + cos⁴x)dx, a, b ∈ R

f'(x) = {– b² + (a – 1)b – 2} + (sin²x + cos⁴x)

Since f(x) is increasing function, so f'(x) ≥ 0

{– b² + (a – 1)b – 2} + sin²x + cos⁴x) ≥ 0

{– b² + (a – 1)b – 2} + 3/4 ≥ 0,

since the minimum

value of sin²x + cos⁴x is 3/4

{– 4b² + 4(a – 1)b – 8 + 3} ≥ 0

–4b² + 4 (a – 1)b – 5 ≥ 0

4b² – 4(a – 1)b + 5 ≤ 0

So its D < 0

16(a – 1)² – 80 < 0

(a – 1)² – 5 < 0

(a – 1)² < 5

|(a – 1)| < √5

–√5 < (a – 1) < √5

1 – √5 < a < 1 + √5

Hence, the value of a is (1 – √5, 1 + √5).