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Chapter 2: Q. 20 (page 197)

Suppose f is a piecewise-defined function, equal to g(x) if x < 2 and h(x) if x 鈮� 2, where g and h are continuous and differentiable everywhere. If g'(2) = h'(2), is the function f necessarily differentiable at x = 2? Why or why not?

Short Answer

Expert verified

Thefunctionfwillalsobedifferentiableatx=2asthelefthandderivativeandrighthandderivativeoffareequalatx=2.

Step by step solution

01

Step 1. Given information is:

f(x)=g(x),ifx<2h(x),ifx鈮�2andg'(2)=h'(2)

02

Step 2. Result

Sinceg'(2)=h'(2)itmeansthatthefunctionsgandharedifferentiableatx=2.Therefore,thefunctionfwillalsobedifferentiableatx=2asthelefthandderivativeandrighthandderivativeoffareequalatx=2.

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