Q108 / 160 · AP EAMCET 2018 (22 Apr Shift 1) · Mathematics

If $f(x)=x^3+a x^2+b x+5 \sin ^2 x$ is an increasing function on $R$, then

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

Solution

$\begin{array}{ll} & \because f^{\prime}(x)>0 \\ \Rightarrow & 3 x^2+2 a x+b+10 \sin x \cos x>0 \\ \Rightarrow & 3 x^2+2 a x+b+5 \sin 2 x>0 \\ \Rightarrow & 3 x^2+2 a x+b-5>0 \\ \Rightarrow \quad & 3 x^2+2 a x+(b-5)>0 \\ \text { Here, } & a>0, a < 0 \\ \therefore & (2 a)^2-4 \times 3 \times(b-5) < 0 \\ & 4 a^2-12(b-5) < 0 \\ & a^2-3(b-5) < 0 \\ & a^2-3 b+15 < 0 .\end{array}$

Hence, the correct answer is option: 'B'.

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