2.a Use Rolle's Theorem to prove that if f ′ ( x ) = 0 for...

2.a Use Rolle's Theorem to prove that if f ′ ( x ) = 0 for all xin an interval ( a , b ), then f is constant on ( a , b ).

b True or False. The product of two increasing functions is increasing. Clarify your answer.

c Find the point on the graph of f ( x ) = 4 − x 2 that is closest to the point ( 0 , 1 ).

Expert Solution

milcah answered 2 years ago

a) use the sequential definition of continuity to prove that f(x)=|x| is continuous. b) theorem 17.3...

a) use the sequential definition of continuity to prove that f(x)=|x| is continuous. b) theorem 17.3 states that if f is continuous at x0, then |f| is continuous at x0. is the converse true? if so, prove it. if not find a counterexample. hint: use counterexample

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1. Prove the Heine-Borel Theorem (Theorem 3.35). 2. Suppose f: X → Y maps from the metric space X to the metric space Y, and x ∈ X. Prove that f is continuous at x if and only if, for any sequence {xn} in X that converges to x, f(xn) → f(x).

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1) A) Determine whether Rolle's Theorem can be applied to f on the closed interval [a,...

1) A) Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = (x − 1)(x − 2)(x − 8), [1, 8] Yes, Rolle's Theorem can be applied. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). No, because f(a) ≠ f(b). B) If Rolle's Theorem can be applied, find all values of c in the...

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