   Chapter 2.2, Problem 62E

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# The left-hand and right-hand derivatives of f at a are defined by f ' − ( a ) = lim h → 0 − f ( a + h ) − f ( a ) h 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?

To determine

(a)To find: f-'4 and f+'4.

Explanation

Formula:

f-'a= lim h  0-fa+h-f(a)h and f+'a= lim h  0+fa+h-f(a)h

Given:

fx=0,                    if x0    5-x,                 if 0<x<415-x                  if x4

Calculation:

The left hand derivative at x = 4 is given by,

f-'4= lim h  0-f4+h-f(4)h

As x approaches  4 from left side, f(x) = 5  x,

f-'4= lim h  0-5-4+h

To determine

(b)To sketch: Graph of f

To determine

(c)To check: The points at which f is discontinuous.

To determine

(d)To check:

The points at which f  is not differentiable.

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