   Chapter 2.8, Problem 64E

Chapter
Section
Textbook Problem

The left-hand and right-hand derivatives off at a are defined by f ' ( a ) = lim h → 0 − f ( a + h ) − f ( a ) a   and   f ' + ( a ) = lim h → 0 + f ( a + h ) − f ( a ) h if these limits exist. Then f'(a) exists if and only if these one-sided derivatives exist and are equal.(a) Find f ' − ( 4 ) and f ' + ( 4 ) for the function f ( x ) = { 0     if   x ≤ 0 5 − x   if   0 < x < 4 1 5 − x   if   x ≥ 4 (b) Sketch the graph of f(c) Where is f discontinuous?(d) Where is f not differentiable?

(a)

To determine

To find: The values of f(4) and f+(4).

Explanation

Result Used: The left-hand and right-hand derivative is of f at x=a

are given by f(a)=limh0f(a+h)f(a)h and f+(a)=limh0+f(a+h)f(a)h.

Given:

The function f(x)={0if x05xif 0<x<415xif x4

Calculation:

Calculate the left-hand derivative of f at x=4.

f(4) =limh0f(4+h)f(4)h

Since h<0, f(4+h)=5(4+h).

f(4) =limh05(4+h)154h =limh054h1h =limh0(hh) =1

Thus, the value of f(4) is 1

(b)

To determine

To sketch: The graph of f(x)={0if x05xif 0<x<415xif x4

(c)

To determine

To find: The points at which f is discontinuous.

(d)

To determine

To find: The points at which f is not differentiable.

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