Intermediate Value Theorem: Calculus

Question

By the intermediate value theorem, the equation x⁴+3x³−x=4 has a solution in which of the following intervals?

 

a. [−1,0]
b. [−2,−1]
c. [1,2]
d. [0,1]

Answer 1

Let f(x) = x⁴+3x³-x-4

 

f(x) is continuous for all x.

 

By the IVT, if f(a) and f(b) have opposite signs, then f(c) = 0 for at least one number c between a and b.

 

f(1) < 0 and f(2) > 0.  So, by the IVT, f(c) = 0 for some number c in the interval (1, 2).