Practicing Success
Let f(x) and g(x) be two functions given by $f(x)=-1+|x-1|,-1 \leq x \leq 3$ and, $g(x)=2-|x+1|,-2 \leq x \leq 2$ Then, |
fog is differentiable at x = -1 and gof is differentiable at x = 1 fog is differentiable at x = -1 and gof is not differentiable at x = 1 fog is differentiable at x = 1 and gof is differentiable at x = -1 none of these |
none of these |
We have, $fog(x)=\left\{\begin{array}{lc} 1+x, & -2 \leq x<-1 \\ -x-1, & -1 \leq x<0 \\ x-1, & 0 \leq x \leq 2 and, $gof(x)=\left\{\begin{array}{lr} 2+x, & -1 \leq x<0 \\ 2-x, & 0 \leq x<1 \\ x, & 1 \leq x<2 \\ 4-x, & 2 \leq x \leq 3 \end{array}\right.$ We observe that (LHD of fog(x) at x = -1) = $\left(\frac{d}{d x}(1+x)\right)_{\text {at } x=-1}=1$ (RHD of fog(x) at x = -1) = $\left(\frac{d}{d x}(-x-1)\right)_{\text {at } x=1}=-1$ Clearly, (LHD of fog(x) at x = -1) ≠ (RHD of fog(x) at x = -1) So, fog is not differentiable at x = -1. Clearly, fog is a polynomial function on [0, 2]. So, it is differentiable at $x=1 \in[0,2]$. Also, (LHD of gof(x) at x = 1) = $\left(\frac{d}{d x}(2-x)\right)_{\text {at } x=1}=-1$ and, (RHD of gof(x) at x = 1) = $\left(\frac{d}{d x}(x)\right)_{\text {at } x=1}=1$. Clearly, these two are not same. So, gof is not differentiable at x = 1. |