8. (10 points) answer each question below.\n(a) (2 points) select an answer choice that correctly completes…

8. (10 points) answer each question below.\n(a) (2 points) select an answer choice that correctly completes the statement:\nthe intermediate value theorem for polynomials states that if ( p(x) ) is a polynor\nand ( p(a) ) and ( p(b) ) have opposite signs, then:\na. ( p(x) ) has no real zero between ( x = a ) and ( x = b )\nb. there is at least one number ( c ) between ( a ) and ( b ) such that ( p(c)=0 )\nc. ( p(x) ) must be increasing on the interval ( (a, b) )\nd. ( p(x) ) has exactly one real zero between ( x = a ) and ( x = b )\n(b) (4 points) let ( f(x)=x^{3}+3 x - 5 ). find ( f(1) ) and ( f(2) ).\nwrite your answer in the box below:\n( f(1)=)\n( f(2)=)

8. (10 points) answer each question below.\n(a) (2 points) select an answer choice that correctly completes the statement:\nthe intermediate value theorem for polynomials states that if ( p(x) ) is a polynor\nand ( p(a) ) and ( p(b) ) have opposite signs, then:\na. ( p(x) ) has no real zero between ( x = a ) and ( x = b )\nb. there is at least one number ( c ) between ( a ) and ( b ) such that ( p(c)=0 )\nc. ( p(x) ) must be increasing on the interval ( (a, b) )\nd. ( p(x) ) has exactly one real zero between ( x = a ) and ( x = b )\n(b) (4 points) let ( f(x)=x^{3}+3 x - 5 ). find ( f(1) ) and ( f(2) ).\nwrite your answer in the box below:\n( f(1)=)\n( f(2)=)

Answer

(a)

Brief Explanations:

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).

Answer:

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

(b)

Explanation:

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:

(f(1)=-1), (f(2)=9)