Let a, b R, b 0, Define a function f(x)= a 2 (x-1), & for x 0 2 x… - Vidyadip

MCQ

Let $\mathrm{a}, \mathrm{b} \in R, \mathrm{b} \neq 0$, Define a function

$f(x)= \begin{cases}\operatorname{a} \sin \frac{\pi}{2}(x-1), & \text { for } x \leq 0 \\ \frac{\tan 2 x-\sin 2 x}{b x^{3}}, & \text { for } x>0\end{cases}$

If $f$ is continuous at $x=0$, then $10-a b$ is equal to ...... .

A
$10$
✓
$14$
C
$8$
D
$3$

✓

Answer

Correct option: B.

$14$

b
$f(x)= \begin{cases}a \sin \frac{\pi}{2}(x-1), & x \leq 0 \\ \frac{\tan 2 x-\sin 2 x}{b x^{3}}, & x>0\end{cases}$

For continuity at $' 0 '$

$\lim _{x \rightarrow 0^{+}} f(x)=f(0)$

$\Rightarrow \lim _{x \rightarrow 0^{+}} \frac{\tan 2 x-\sin 2 x}{b x^{3}}=-a$

$\Rightarrow \lim _{x \rightarrow 0^{+}} \frac{\frac{8 x^{3}}{3}+\frac{8 x^{3}}{3 !}}{b x^{3}}=-a$

$\Rightarrow 8\left(\frac{1}{3}+\frac{1}{3 !}\right)=-a b$

$\Rightarrow 4=-a b$

$\Rightarrow 10-a b=14$

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