If the function f(x) = { 3ax + b, if x > 1; 11, if x = 1; 5ax - 2b, if x < 1 } is continuous at x = 1, find 'a' and 'b'.

Steps to Solve

1. Identify continuity condition For the function to be continuous at $x = 1$, the left-hand limit, right-hand limit, and the function value at that point must all be equal:

$$ \lim_{x \to 1^-} f(x) = f(1) = \lim_{x \to 1^+} f(x) $$

2. Calculate the left-hand limit We find the limit of ( f(x) ) as ( x ) approaches 1 from the left:

$$ \lim_{x \to 1^-} f(x) = 5a(1) - 2b = 5a - 2b $$

3. Calculate the function value at x = 1 The function value at ( x = 1 ) is:

$$ f(1) = 11 $$

4. Calculate the right-hand limit We find the limit of ( f(x) ) as ( x ) approaches 1 from the right:

$$ \lim_{x \to 1^+} f(x) = 3a(1) + b = 3a + b $$

5. Set up the equations for continuity We set the left-hand limit equal to the function value and the right-hand limit equal to the function value:

6. ( 5a - 2b = 11 ) (left limit = function value)

7. ( 3a + b = 11 ) (right limit = function value)

8. Solve the system of equations Now we have a system of equations:

9. ( 5a - 2b = 11 )

10. ( 3a + b = 11 )

To solve, we can express ( b ) from the second equation:

$$ b = 11 - 3a $$

Substitute ( b ) into the first equation:

$$ 5a - 2(11 - 3a) = 11 $$

Simplify and solve for ( a ):

$$ 5a - 22 + 6a = 11 $$

Combine terms:

$$ 11a - 22 = 11 $$

Add 22 to both sides:

$$ 11a = 33 $$

Divide by 11:

$$ a = 3 $$

7. Find value of b Substitute ( a = 3 ) back into the equation for ( b ):

$$ b = 11 - 3(3) $$

This simplifies to:

$$ b = 11 - 9 = 2 $$