The function $$f(x)=x2$$−$$1x2$$−$$3x+2+$$$$cos$$⁡$$($$|x|$$)$$ is not differentiable at:

Question

The function $$f(x)=x2$$−$$1x2$$−$$3x+2+$$$$cos$$⁡$$($$|x|$$)$$ is not differentiable at:

Infinity Learn · Accepted Answer

We  have $$x2$$−$$3x+2=x$$−$$1+x$$−$$2$$                                        $$=1$$−$$x2$$−x,    if x≤$$1x$$−$$12$$−x,    if $$12This$$ function is differentiable  at  all points  except possible  at $$x=1$$ and $$x=2$$.              $$Lf(1)$$  $$=limx$$→$$1$$−$$f(x)$$−$$f(1)x$$−$$1$$                             $$=limx$$→$$1$$−$$x2$$−$$1x$$−$$2+cosx$$−$$cos1x$$−$$1$$                            $$=$$  $$0$$−$$1sin1=$$−$$sin1$$.and  $$Rf(1)$$  $$=$$ limx→$$1+f(x)$$−$$f(1)x$$−$$1$$                          $$=limx$$→$$1+$$−$$x2$$−$$1x$$−$$2+cosx$$−$$cos1x$$−$$1$$                         $$=$$ $$0$$−$$sin1=$$−$$sin1$$∴       f is differentiable at $$x=1$$.        $$Lf(2)=limx$$→$$2$$−$$f(x)$$−$$f(2)x$$−$$2$$                      $$=$$ limx→$$2$$−−$$x2$$−$$1x$$−$$1+cosx$$−$$cos2x$$−$$2$$                     $$=$$ −$$4$$−$$12$$−$$1$$−$$sin2=$$−$$3$$−$$sin2$$    $$Rf(2)=$$ limx→$$2+f(x)$$−$$f(2)x$$−$$2$$                   $$=$$ limx→$$2+x2$$−$$1x$$−$$1+cosx$$−$$cos2x$$−$$2$$                  $$=$$ $$22$$−$$12$$−$$1$$−$$sin2=3$$−$$sin2As$$ $$Lf(2)$$≠$$Rf(2)$$, f is not differentiable at $$x=2$$.Hence (4) is the correct answer.

The function f(x)=x2−1x2−3x+2+cos⁡(|x|) is not differentiable at:

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