MCQ
Let $\text{f}(\text{x})=\text{x}^3+\text{a}\text{x}^2+\text{b}\text{x}+5\sin^2\text{x}$ be an increasing function on R. Then, a and b satisfy:
- A$a^2 - 3b - 15 > 0$
- B$a^2 - 3b + 15 > 0$
- ✓$a^2 - 3b + 15 < 0$
- D$a < 0$ and $b > 0$
✓
Answer
Correct option: C.
$a^2 - 3b + 15 < 0$
$\text{f}(\text{x})=\text{x}^3+\text{a}\text{x}^2+\text{b}\text{x}+5\sin^2\text{x}$
$\text{f}'(\text{x})=3\text{x}^2+2\text{a}\text{x}+(\text{b}+5\sin2\text{x})$
Given, f(x) is increasing on R.
$\Rightarrow\text{f}'(\text{x})>0,\forall\ \text{x}\in\text{R}$
$\Rightarrow3\text{x}^2+2\text{a}\text{x}+(\text{b}+5\sin2\text{x}) > 0,\forall\ \text{x}\in\text{R}$
Since, the quadratic function is $ > 0$, its discriminant is $ < 0$.
$\Rightarrow(2\text{a})^2-4(3)(\text{b}+5\sin2\text{x}) < 0$
$\Rightarrow4\text{a}^2-12\text{b}-60\sin2\text{x} < 0$
$\Rightarrow\text{a}^2-3\text{b}-15\sin2\text{x} < 0$
We know that the minimum value of $\sin2\text{x}$ is $-1$.
$\therefore\ \text{a}^2-3\text{b}-15 < 0$
$\text{f}'(\text{x})=3\text{x}^2+2\text{a}\text{x}+(\text{b}+5\sin2\text{x})$
Given, f(x) is increasing on R.
$\Rightarrow\text{f}'(\text{x})>0,\forall\ \text{x}\in\text{R}$
$\Rightarrow3\text{x}^2+2\text{a}\text{x}+(\text{b}+5\sin2\text{x}) > 0,\forall\ \text{x}\in\text{R}$
Since, the quadratic function is $ > 0$, its discriminant is $ < 0$.
$\Rightarrow(2\text{a})^2-4(3)(\text{b}+5\sin2\text{x}) < 0$
$\Rightarrow4\text{a}^2-12\text{b}-60\sin2\text{x} < 0$
$\Rightarrow\text{a}^2-3\text{b}-15\sin2\text{x} < 0$
We know that the minimum value of $\sin2\text{x}$ is $-1$.
$\therefore\ \text{a}^2-3\text{b}-15 < 0$
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