Question

Let $F(x) = x^3 + ax^2 + bx + 5 sin^2\, x$ be an increasing function in the set of real number $R$. Then a and b satisfy the condition.

• A. $a^2 - 3b - 15 > 0$
• B. $a^2 - 3b + 15 > 0$
• C. $a^2 + 3b - 15 < 0$
• D. $a > 0$ and $b > 0$

Answer 1

Answer: (C)

We have $f(x) = x^3 + ax^2 + bx + 5 sin^2\, x$
$\Rightarrow f '(x) 3x^2 + 2ax + b + 5\,sin\,2x$
$\because f(x)$ is an increasing function
$\therefore f '\left(x\right) >0 \Rightarrow 3x^{2}+2ax+b+5\,sin\,2x > 0,$
$\left(\because sin\,2x < 1\right)$
$\therefore 0 < 3x^{2}+2ax+b+5\,sin\,2x < 3x^{2} + 2ax+b+5$
$\Rightarrow 3x^{2}+2ax+b+5 > 0$
$\Rightarrow 4a^{2}+4.3\left(b+5\right) < 0 \Rightarrow a^{2}+3b-15 < 0$
[$\because ax^{2}+bx+c > 0$ or all real $x$ if . $a > 0$ and discriminant $< 0$]