   Chapter 2.6, Problem 59E

Chapter
Section
Textbook Problem

Let f ( x ) = { x 2  if   x ≤ 1 ax + b   i f   x > 1 Find the values of a and b such that f is continuous and has a derivative at x = 1. Sketch the graph of f.

To determine

To sketch: The value of a and b such that the given function is continuous and has a derivative at x=1 and to sketch the graph of the given function.

Explanation

Given information:

The function is f(x)={x2 if x1ax+b if x>1 (1)

Calculation:

Section1:

Apply the condition for continuity of function.

limx1f(x)=limx1+f(x)=f(1) .

Left hand limit of the given function is,

limx1f(x)=limx1x2=1 (2)

Right hand limit of the given function is,

limx1+f(x)=limx1+(ax+b)=a+b (3)

From equation (2) and (3).

a+b=1 (4)

Apply the condition of differentiability at x=1 in equation (1).

The function to be differentiable at x=1 , gives a=2 .

Substitute 2 for a in equation (4) and get the value of b.

2+b=1b=1

Thus, the value of a and b is 2 and 1 respectively

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