The polynomial function is continuous for all real

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

The polynomial function

p(x)=x^3-4x-1

is continuous for all real

x

. Which of the following justifies whether the equation

p(x)=2

has at least one real solution in the interval

(1,3)

? (a) Yes;

p

is continuous on

[1,3]

and

p'(1)<2<p'(3)

. (b) No; both

p(1)

and

p(3)

are greater than

2

, so the equation has no solution in

(1,3)

. (c) Yes;

p

is continuous on

[1,3]

and

p(1)<2<p(3)

. (d) Yes;

p(2)=1<2

, so the equation must have a solution near

x=2