Question

Let function $f\left(x\right)=\{\begin{array}{c}\cos x+2;x\le 0\\ A\sin x+B;x>0\end{array}$
is continuous everywhere but not differentiable $(A,B\in R)$, then

• A. $A=0$
• B. $B\ne 3$
• C. $A+B=3$
• D. $B-A\ne 3$

Answer 1

The correct option is D $B-A\ne 3$

For continuity $f\left(0^{-}\right)=f\left(0^{+}\right)=f\left(0\right)$
$\Rightarrow 3=B$
and for non-differentiability
$f^{\prime }\left(0^{-}\right)\ne f^{\prime }\left(0^{+}\right)$
$\Rightarrow \underset{x\to 0^{-}}{lim}(-\sin x)\ne \underset{x\to 0^{+}}{lim}\left(A\cos x\right)\Rightarrow A\ne 0$
$\Rightarrow B-A\ne 3$