Find the values of a and b that make the function f(x) continuous at every where

Question

Find the values of 'a' and 'b', if possible, that will make the following function f(x) continuous everywhere:

f(x) =

X² +5, if x > 2

a(x+1)+b, if - 1<x<=2

2x^3+x+7, if x < =-1.

Answer 1

f(x) =

X² +5, if x > 2

a(x+1)+b, if - 1<x<=2

2x^3+x+7, if x < =-1.

To be continuous at x=2,

f(2) = a(2+1)+b = 2²+5

3a + b = 9

To be continuous at x=-1,

f(-1) = a(-1+1) + b = 2(-1)³ + (-1) + 7

b = 4

Substituting, 3a + 4 = 9

a = 5/3

Check:

Does (5/3)(2+1)+4 = 9 ?

5+4 = 9 ?

9 = 9 ? yes



Does (5/3)(-1+1) + 4 = 2(-1) + (-1) + 7 ?

0 + 4 = 4 ? yes