Question

1. (10 points) answer each question below.

(a) (2 points) select an answer choice that correctly completes the statement:
the intermediate value theorem for polynomials states that if ( p(x) ) is a polynor
and ( p(a) ) and ( p(b) ) have opposite signs, then:
a. ( p(x) ) has no real zero between ( x = a ) and ( x = b )
b. there is at least one number ( c ) between ( a ) and ( b ) such that ( p(c)=0 )
c. ( p(x) ) must be increasing on the interval ( (a, b) )
d. ( p(x) ) has exactly one real zero between ( x = a ) and ( x = b )
(b) (4 points) let ( f(x)=x^{3}+3 x - 5 ). find ( f(1) ) and ( f(2) ).
write your answer in the box below:
( f(1)=)
( f(2)=)

Explanation:

(a)

The Intermediate Value Theorem for Polynomials states that for a continuous function (and polynomials are continuous everywhere), if \(P(a)\) and \(P(b)\) have opposite signs, then there is at least one \(c\) in the interval \((a,b)\) such that \(P(c)=0\). Option A is incorrect as it contradicts the theorem. Option C is wrong because the theorem doesn't say anything about the function being increasing. Option D is incorrect as it could have more than one zero (the theorem just guarantees at least one).

Step 1: Find \(f(1)\)

Substitute \(x = 1\) into \(f(x)=x^{3}+3x - 5\)
\(f(1)=1^{3}+3\times1 - 5\)
\(=1 + 3-5\)
\(=-1\)

Step 2: Find \(f(2)\)

Substitute \(x = 2\) into \(f(x)=x^{3}+3x - 5\)
\(f(2)=2^{3}+3\times2 - 5\)
\(=8+6 - 5\)
\(=9\)

Answer:

B. There is at least one number \(c\) between \(a\) and \(b\) such that \(P(c) = 0\)

(b)