Question
Question: Let ƒ(x) = x<sup>3</sup> + ax<sup>2</sup> + bx + 5 sin<sup>2</sup>x be an increasing function in the...
Let ƒ(x) = x3 + ax2 + bx + 5 sin2x be an increasing function in the set of real numbers R. Then a and b satisfy the condition –
A
a2 – 3b – 15 > 0
B
a2 – 3b + 15 > 0
C
a2 – 3b – 15 < 0
D
a > 0 and b > 0
Answer
a2 – 3b – 15 < 0
Explanation
Solution
We have ƒ(x) = x3 + ax2 + bx + 5 sin2 x
̃ ƒ¢(x) = 3x2 + 2ax + b + 5 sin 2x
Q ƒ (x) is an increasing function
\ ƒ¢(x) > 0 ̃ 3x2 + 2ax + b + 5 sin 2x > 0,
Q sin 2x < 1
\ 0 < 3x2 + 2ax + b + 5 sin 2x < 3x2 + 2ax + b + 5
̃ 3x2 + 2ax + b + 5 > 0
̃ 4a2 – 4.3 (b + 5) > 0
̃ a2 – 3b – 15 < 0.