Question

a. use the intermediate value theorem to show that the following equation has a solution on the given interval. x³ - 5x² + 2x=-2; (-1,5) b. use the graphing utility to find all the solutions to the equation on the given interval. c. illustrate your answers with an appropriate graph. the intermediate value theorem states that if f is continuous on the interval a,b and l is a number strictly between f(a) and f(b), then there exists at least one number c in (a,b) satisfying f(c)=l. for which values of x is the function f(x)=x³ - 5x² + 2x continuous? a. it is continuous on -1,5 but not for all x b. it is continuous for all x c. it is continuous for some x, but not on -1,5 d. it is not continuous on any interval

Explanation:

Step1: Recall polynomial - continuity property

Polynomial functions are continuous everywhere. The function \(f(x)=x^{3}-5x^{2}+2x\) is a polynomial function of degree 3.

Step2: Analyze continuity on the interval \([-1,5]\)

Since polynomial functions are continuous for all real - valued \(x\), \(f(x)\) is continuous on the interval \([-1,5]\) and for all \(x\) in general.

Answer:

B. It is continuous for all \(x\)