Assertion (A): Let f(x)=x^3+a x^2+b x+5 ^2 x, then the condition … - Vidyadip

MCQ

Assertion $(A):$ Let $f(x)=x^3+a x^2+b x+5 \sin ^2 x,$ then the condition that $f(x)$ is always one $-$ one function is $a^2-3 b+15 < 0$.
Reason $(R) : f(x)$ to be one one either $f$ is strictly increasing or strictly decreasing.

✓
Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$
B
Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A).$
C
$(A)$ is true but $(R)$ is false.
D
$(A)$ is false but $(R)$ is true.

✓

Answer

Correct option: A.

Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$

$ f(x)=x^3+a x^2+b x+5 \sin ^2 x$
$\therefore f^{\prime}(x)=3 x^2+2 a x+b+5 \sin 2 x$
For one $-$ one function, $f^{\prime}(x)>0$ for $x \in R$
$\Rightarrow 3 x^2+2 a x+b+5 \sin 2 x>0$
$\Rightarrow 3 x^2+2 a x+(b-5)>0$
$ \Rightarrow(2 a)^2-4 \cdot 3(b-5)<0$
$\Rightarrow a^2-3 b+15<0$

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