Determine the values of a and b which would result in the function f(x) being differentiable at x=2.

Question

f(x) = { 2ax + 10 x<or equal to 2

{ bx^2 - 4x + 9 x>2

Answer 1

Question:

Determine the values of a and b which would result in the function f(x) being differentiable at x=2.

f(x) = { 2ax + 10 x<or equal to 2

{ bx^2 - 4x + 9 x>2

Solution:

The trick here is to get both the function and it's derivative to be the same at the point x = 2.

Equation #1:

f1(x) = f2(x)

2ax + 10 = bx^2 - 4x + 9

bx^2 - (4x + 2ax) + (9 - 10) = 0

bx^2 - (2a + 4)x - 1 = 0

Equation #2:

f1'(x) = f2'(x)

2a = 2bx - 4

a = bx - 2

We now have 2 equations with 2 unknowns.

bx^2 - (2a + 4)x - 1 = 0

a = bx - 2

Substitute for 'a':

bx^2 - (2(bx - 2) + 4)x - 1 = 0

bx^2 - (2bx - 4 + 4)x - 1 = 0

bx^2 - 2bx^2 - 1 = 0

-bx^2 - 1 = 0

bx^2 + 1 = 0

bx^2 = - 1

b = -1 / x^2

Since x = 2,

b = -1 / 2^2

b = -1/4

a = bx - 2

a = (-1/4)(2) - 2

a = -1/2 - 2

a = -5/2

THEREFORE:

a = -5/2, b = -1/4 make f(x) differentiable at x = 2.

f(x) = { -5x + 10 for x <= 2

{ (-1/4)*x^2 - 4x + 9 for x > 2

Answer 2

the function needs to be continuous for it to be differentiable, so at x=2 the equations must equal:

2a(2)+10=b(2²)-4*2+9, 4a+10=4b+1, 4a=4b-9, a=b-9/4

now we take the derivative of the function at x=2 because the derivatives need to be the same:

2a=2b(2)-4, a=2b-2

now use these 2 equations to solve for a and b:

2b-2=b-9/4, b=-1/4, a=-1/4-9/4, a=-10/4