The function g (x) = [ beginmatrixx+b, &x

Question

The function g (x) = [ beginmatrixx+b, &x< 0 cos x,&x ge0 endmatrix can be made differentiable at x = 0, (A) if b is equal to zero (B) if b is not equal to zero (C) if b takes any real value (D) for no value of b

Tardigrade Admin · Accepted Answer

First f(x) must be continuous at x=0, for which 0+b= cos 0 or b=1 Also g'(x) = [ beginmatrix1, x< 0 -sin x,&x > 0 endmatrix ⇒ g'(0+)=0 and g'(0-)=1 Hence, f(x) is non-differentiable at x=0. Thus g cannot be made differentiable for any value of b

Q. The function $g (x) = [x + b, cos x, x < 0 x \geq 0$ can be made differentiable at x = 0,

if b is equal to zero

if b is not equal to zero

if b takes any real value

for no value of b

First $f (x)$ must be continuous at $x = 0,$ for which $0 + b = cos 0$ or $b = 1$
Also $g^{'} (x) = [1, - s in x, x < 0 x > 0$
$\Rightarrow g^{'} (0^{+}) = 0$ and $g^{'} (0^{-}) = 1$
Hence, $f (x)$ is non-differentiable at $x = 0$ .
Thus $g$ cannot be made differentiable for any value of $b$

Q. The function g(x)=[x+b,cosx,​x<0x≥0​ can be made differentiable at x = 0,