if f(x)=x³ - 4x + 6, show that there are values c for which f(c) equals (a) π, (b) -√3, and (c) 5,000,000…

if f(x)=x³ - 4x + 6, show that there are values c for which f(c) equals (a) π, (b) -√3, and (c) 5,000,000. (a) the function f is a polynomial, which is everywhere - continuous. what is the best way to begin the problem? a. find values of x₁ and x₂ for which f(x₁)≤π≤f(x₂) or f(x₁)≥π≥f(x₂). b. find values of x₁ and x₂ for which f(x₁)≤0≤f(x₂) or f(x₁)≥0≥f(x₂). c. find values of x₁ and x₂ for which f(x₁)=0 or f(x₂)=0. d. find values of x₁ and x₂ for which f(x₁)≤π and f(x₂)≤π or f(x₁)≥π and f(x₂)≥π.

if f(x)=x³ - 4x + 6, show that there are values c for which f(c) equals (a) π, (b) -√3, and (c) 5,000,000. (a) the function f is a polynomial, which is everywhere - continuous. what is the best way to begin the problem? a. find values of x₁ and x₂ for which f(x₁)≤π≤f(x₂) or f(x₁)≥π≥f(x₂). b. find values of x₁ and x₂ for which f(x₁)≤0≤f(x₂) or f(x₁)≥0≥f(x₂). c. find values of x₁ and x₂ for which f(x₁)=0 or f(x₂)=0. d. find values of x₁ and x₂ for which f(x₁)≤π and f(x₂)≤π or f(x₁)≥π and f(x₂)≥π.

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). For a continuous function (f(x)) and a target value (k), we need to find (x_1) and (x_2) such that (k) lies between (f(x_1)) and (f(x_2)). Here (k = \pi), so we need to find (x_1) and (x_2) for which (f(x_1)\leq\pi\leq f(x_2)) or (f(x_1)\geq\pi\geq f(x_2)).

Answer:

A. Find values of (x_1) and (x_2) for which (f(x_1)\leq\pi\leq f(x_2)) or (f(x_1)\geq\pi\geq f(x_2)).