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Question

Let fx=1xfor x1ax2+bfor x<1

If f (x) is continuous and differentiable at any point, then
(a) a=12, b=-32
(b) a=-12, b=32
(c) a = 1, b = − 1
(d) none of these


Solution

(b) a=-12, b=32

We have,fx=-1x,       x-1ax2+b,    -1<x<11x,          x1Given: fx is differentiable and continuous at every point.Consider a point x=1limx1-fx=limx1+fxlimx1-ax2+b=limx1+1xa+b=1          ...iIt is also differentiable at x=1limx1-fx-f1x-1=limx1+fx-f1x-1limx1-ax2+b-1x-1=limx1+1x-1x-1limx1-ax2-ax-1=limx1+1-xx-1x       Using ilimx1-ax+1=limx1+-x2a=-1a=-12Plugging a=-12 in i we get,b=32a=-12, b=32

Mathematics
RD Sharma XII Vol 1 (2015)
Standard XII

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