State Intermediate -Value Theorem. Suppose f is continuous function on theinterval [a, b] where a < b.Show that there is at least one number c in theinterval [a, b] such that f (c) = c.

Let 0 < a < b and f : [a, b] → R be continuous on [a, b]. Prove that there exists c ∈ [a, b] suchthatZ batf (t)dt = 12 f (c)(b2 − a2).Hint) Consider the functionh(x) = 12 f (x)(b2 − a2) −Z batf (t)dt.Use the comparison principle and the intermediate value theorem

State the intermediate value theorem.(b.) Prove that there is some number x such that x2 + x − 1 = 0.

According to the intermediate value theorem, if you have a function where f(4) = 5 and f(6) = 3, there will be at least one point x between 4 and 6 where f(x) = 4.

Suppose that f : [a, b] → R is a continuous function that satisfies f (x) ∈ [a, b]for all x ∈ [a, b]. Prove that if f is differentiable on (a, b) and |f ′(x)| < 1 for all x ∈ (a, b),then f has exactly one fixed point. (Hint: We know that there is at least one fixed point byHomework 6. Try a proof by contradiction, and use the Mean Value Theorem.

Let f be a function such that f (1) = − 2 and f (5) = 7. Which of the following conditions ensures that f(c) = 0 for some value c in the open interval (1,5)?Responses∫15𝑓(𝑥)𝑑𝑥 exists. Integration of f of x of d x exists from 1 to 5f is increasing on the closed interval [1, 5].f is increasing on the closed interval [1, 5].f is continuous on the closed interval [1, 5].f is continuous on the closed interval [1, 5].f is defined for all values of x in the closed interval [1, 5].