Practicing Success
If the function f(x) given by $f(x)= \begin{cases}2^{1 /(x-1)}, & x<1 \\ a x^2+b x, & x \geq 1\end{cases}$ is everywhere differentiable, then |
a = 0, b = 1 a = 0, b = 0 a = 1, b = 0 none of these |
a = 0, b = 0 |
Clearly, f(x) is everywhere continuous and differentiable except possibly at x = 1. At x = 1, we have $\lim\limits_{x \rightarrow 1^{-}} f(x)=\lim\limits_{x \rightarrow 1^{-}} 2^{1 /(x-1)}=\lim\limits_{h \rightarrow 0} 2^{-1 / h}=0$ $\lim\limits_{x \rightarrow 1^{+}} f(x)=\lim\limits_{x \rightarrow 1^{+}}\left(a x^2+b x\right)=a+b$ and, $f(1)=a+b$ For f(x) to be differentiable at x = 1, it must be continuous there at. ∴ $\lim\limits_{x \rightarrow 1^{-}} f(x)=\lim\limits_{x \rightarrow 1^{+}} f(x)=f(1)$ ⇒ 0 = a + b .........(i) Also, (LHD at x = 1) = $\lim\limits_{x \rightarrow 1^{-}} \frac{f(x)-f(1)}{x-1}$ ⇒ (LHD at x = 1) = $\lim\limits_{h \rightarrow 0} \frac{f(1-h)-f(1)}{-h}$ ⇒ (LHD at x = 1) = $\lim\limits_{h \rightarrow 0} \frac{2^{-1 / h}-(a+b)}{-h}$ ⇒ (LHD at x = 1) = $\lim\limits_{h \rightarrow 0} \frac{2^{-1 / h}}{-h}$ [∵ a + b = 0] ⇒ (LHD at x = 1) = $-\lim\limits_{h \rightarrow 0} \frac{1 / h}{2^{1 / h}}$ ${\left[\frac{\infty}{\infty} \text { form }\right]}$ ⇒ (LHD at x = 1) = $-\lim\limits_{h \rightarrow 0} \frac{-1 / h^2}{2^{1 / h} \log _e 2\left(-\frac{1}{h^2}\right)}$ ⇒ (LHD at x = 1) = $-\lim\limits_{h \rightarrow 0} \frac{2^{-1 / h}}{\log _e 2}=\frac{0}{\log _e 2}=0$ and, ⇒ (RHD at x = 1) = $\lim\limits_{x \rightarrow 1^{+}} \frac{f(x)-f(1)}{x-1}$ ⇒ (RHD at x = 1) = $\lim\limits_{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}$ ⇒ (RHD at x = 1) = $\lim\limits_{h \rightarrow 0} \frac{a(1+h)^2+b(1+h)-(a+b)}{h}$ ⇒ (RHD at x = 1) = $\lim\limits_{h \rightarrow 0} \frac{a h^2+2 a h+b h}{h}$ ⇒ (RHD at x = 1) = $\lim\limits_{h \rightarrow 0} a h+2 a+b=2 a+b$ For f(x) to be differentiable at x = 1, we must have (LHD at x = 1) = (RHD at x = 1) ⇒ 0 = 2a + b .............(ii) Solving (i) and (ii), we get a = b = 0. |