using the intermediate value theorem, show that the function f has a zero between a and b. f(x)=x³ + 3x²…

using the intermediate value theorem, show that the function f has a zero between a and b. f(x)=x³ + 3x² - 9x - 13; a = -5, b = -4 what is f(-5)? -18 what is f(-4)? 7 choose the correct statement below that explains why the given polynomial has a zero between -5 and -4, according to the intermediate value theorem. a. since f(-5) and f(-4) are opposite in sign, there exists at least one zero between -5 and -4. b. since f(-4) is greater than f(-5), the function is increasing and so there must be one real zero between -5 and -4.

using the intermediate value theorem, show that the function f has a zero between a and b. f(x)=x³ + 3x² - 9x - 13; a = -5, b = -4 what is f(-5)? -18 what is f(-4)? 7 choose the correct statement below that explains why the given polynomial has a zero between -5 and -4, according to the intermediate value theorem. a. since f(-5) and f(-4) are opposite in sign, there exists at least one zero between -5 and -4. b. since f(-4) is greater than f(-5), the function is increasing and so there must be one real zero between -5 and -4.

Answer

Explanation:

Step1: Recall Intermediate - Value Theorem

The Intermediate - Value Theorem states that if a function (y = f(x)) is continuous on a closed interval ([a,b]), and (k) is a number between (f(a)) and (f(b)), then there exists at least one number (c) in the interval ((a,b)) such that (f(c)=k). In the case of finding a zero, if (f(a)) and (f(b)) have opposite signs, then there is at least one (c\in(a,b)) such that (f(c) = 0).

Step2: Evaluate (f(-5)) and (f(-4))

We are given (f(x)=x^{3}+3x^{2}-9x - 13), (a=-5), (b = - 4). [ \begin{align*} f(-5)&=(-5)^{3}+3\times(-5)^{2}-9\times(-5)-13\ &=-125 + 3\times25+45-13\ &=-125+75 + 45-13\ &=-18 \end{align*} ] [ \begin{align*} f(-4)&=(-4)^{3}+3\times(-4)^{2}-9\times(-4)-13\ &=-64+3\times16 + 36-13\ &=-64 + 48+36-13\ &=7 \end{align*} ]

Step3: Apply the Intermediate - Value Theorem

Since (f(-5)=-18) and (f(-4)=7), (f(-5)) and (f(-4)) have opposite signs.

Answer:

A. Since (f(-5)) and (f(-4)) are opposite in sign, there exists at least one zero between (-5) and (-4).