Answer to Question #258992 in Calculus for Pankaj

Question #258992

Is the function f : R ➡R, defined by f(x) = 1-|x| is differentiable at x=1

1
Expert's answer
2021-11-02T18:36:27-0400
limh0f(1+h)f(1)h=limh011+h1+1h\lim\limits_{h\to0^-}\dfrac{f(1+h)-f(1)}{h}=\lim\limits_{h\to0^-}\dfrac{1-|1+h|-1+|1|}{h}

=limh0hh=1=\lim\limits_{h\to0^-}\dfrac{-h}{h}=-1



limh0+f(1+h)f(1)h=limh0+11+h1+1h\lim\limits_{h\to0^+}\dfrac{f(1+h)-f(1)}{h}=\lim\limits_{h\to0^+}\dfrac{1-|1+h|-1+|1|}{h}

=limh0+hh=1=\lim\limits_{h\to0^+}\dfrac{-h}{h}=-1

limh0f(1+h)f(1)h=1=limh0+f(1+h)f(1)h\lim\limits_{h\to0^-}\dfrac{f(1+h)-f(1)}{h}=-1=\lim\limits_{h\to0^+}\dfrac{f(1+h)-f(1)}{h}

Then


limh0f(1+h)f(1)h=1\lim\limits_{h\to0}\dfrac{f(1+h)-f(1)}{h}=-1

Therefore the function f(x)=1xf(x) = 1-|x| is differentiable at x=1x=1 and f(1)=1.f'(1)=-1.


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