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Question

Function f(x) is defined as f(x)=x12x27x+5, x113, x=1
Is f(x) differentiable at x=1 if yes find f(1).

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Solution

We have,
f(x)=x12x27x+5,x113,x=1
limx1(x1)(x1)(2x5)=13
As f:x is continuous.
f(1+)=limx0f(1+h)f(1)x=h!h(2h+3)+13h
limx03+2h33h(2h3)=23(1)=29f(1)=limx0f(1)f(1h)h=limh013132hh=13+13+2hh=32h+33(3+2h)h=limx323(3+2h)
=29
Therefore f(x) is differentiable f(1)=29.

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