If f(x) is continuous and differentiable and f(x) = ax^4 + 5x; x<=2 and bx^2 - 3x; x> 2, then how do you find b?

1 Answer
Nov 7, 2016

b = 15/2

Explanation:

As f(x) is continuous at x=2, we have

lim_(x->2^-)f(x) = lim_(x->2^+)f(x)

=> a(2^4)+5(2) = b(2^2)-3(2)

=> 16a+10 = 4b-6

=> a = 1/4b - 1

As f(x) is differentiable at x=2, the limit f'(2) = lim_(x->2)(f(x)-f(2))/(x-2) must exist. We can tell what the one sided limits will evaluate to by calculating the derivatives of the components of the piecewise defined functions on either side of 2.

lim_(x->2^-)(f(x)-f(2))/(x-2) = lim_(x->2^+)(f(x)-f(2))/(x-2)

=> 4a(2^3)+5 = 2b(2)-3

=> 32a+5 = 4b-3

Substituting in a = 1/4b-1, we have

32(1/4b-1) + 5 = 4b-3

=> 8b - 27 = 4b - 3

=> 4b = 30

:. b = 15/2