Consider the function. f(x)= cc a (7 x-12-x^2 ) b |x^2-7 x+12 | &… - Vidyadip

MCQ

Consider the function. $f(x)=\left\{\begin{array}{cc} \frac{a\left(7 x-12-x^2\right)}{b\left|x^2-7 x+12\right|} & , x<3 \\ 2^{\frac{\sin (x-3)}{x-[x]}} & , x>3 \\ b & , x=3 \end{array}\right. $
Where $[ x ]$ denotes the greatest integer less than or equal to $x$. If $S$ denotes the set of all ordered pairs $(a, b)$ such that $f(x)$ is continuous at $x=3$, then the number of elements in $S$ is :

A
2
B
Infinitely many
C
4
D
1

✓

Answer

$f\left(3^{-}\right)=\frac{a}{b} \frac{\left(7 x-12-x^2\right)}{\left|x^2-7 x+12\right|} ($ for $f(x)$ to be cont.$)$
$f\left(3^{-}\right)=\frac{-a}{b} \frac{(x-3)(x-4)}{(x-3)(x-4)} ; x<3 \Rightarrow \frac{-a}{b}$
Hence $f\left(3^{-}\right)=\frac{-a}{b}$
Then $f\left(3^{+}\right)=2^{\lim _{x \rightarrow+}\left(\frac{\sin (x-3)}{x-3}\right)}=2$ and $f(3)=b$.
Hence $f(3)=f\left(3^{+}\right)=f\left(3^{-}\right)$
$\Rightarrow b=2=-\frac{a}{b}$
$b=2, a=-4$
Hence only $1$ ordered pair $(-4,2)$.

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