watch the video and then solve the problem given below. click here to watch the video. using the…

watch the video and then solve the problem given below. click here to watch the video. 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

watch the video and then solve the problem given below. click here to watch the video. 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

Answer

Explanation:

Step1: Evaluate (f(a))

Substitute (a = - 5) into (f(x)=x^{3}+3x^{2}-9x - 13) [ \begin{align*} f(-5)&=(-5)^{3}+3(-5)^{2}-9(-5)-13\ &=-125 + 75+45-13\ &=-18 \end{align*} ]

Step2: Evaluate (f(b))

Substitute (b=-4) into (f(x)=x^{3}+3x^{2}-9x - 13) [ \begin{align*} f(-4)&=(-4)^{3}+3(-4)^{2}-9(-4)-13\ &=-64+48 + 36-13\ &=7 \end{align*} ]

Step3: Apply the Intermediate Value Theorem

Since (f(x)) is a polynomial, it is continuous on the interval ([-5,-4]). Also, (f(-5)=-18<0) and (f(-4) = 7>0). By the Intermediate Value Theorem, for (N = 0) (a value between (f(-5)) and (f(-4))), there exists a number (c\in(-5,-4)) such that (f(c)=0)

Answer:

Since (f(-5)=-18<0) and (f(-4)=7>0) and (f(x)) is continuous (as a polynomial), by the Intermediate Value Theorem, there is a zero of (f(x)) between (a=-5) and (b = - 4)