Suppose f : [a, b] −→ R is continuous on [a, b] and that f attains...

Suppose f : [a, b] −→ R is continuous on [a, b] and that f attains its minimum value at ξ ∈ (a, b). If f ' (ξ) exists, prove that f ' (ξ) = 0.

Expert Solution

Colby Messinger answered 5 months ago

a) Suppose f:[a, b] → R is continuous on [a, b] and differentiable on (a, b)...

a) Suppose f:[a, b] → R is continuous on [a, b] and differentiable on (a, b) and f ' < -1 on (a, b). Prove that f is strictly decreasing on [a, b]. b) Suppose f:[a, b] → R is continuous on [a, b] and differentiable on (a, b) and f ' ≠ -1 on (a, b). Why must it be true that either f ' > -1 on all of (a, b) or f ' < -1 on all...

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Suppose a function f : R → R is continuous with f(0) = 1. Show that...

Suppose a function f : R → R is continuous with f(0) = 1. Show that if there is a positive number x0 for which f(x0) = 0, then there is a smallest positive number p for which f(p) = 0. (Hint: Consider the set {x | x > 0, f(x) = 0}.)

Assigned Exercise IX.1. IX.1. (a) Suppose that f : [a, b] → R is continuous. Define...

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Integral Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information...

Integral Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information to prove thatL(f)≤L(g)andU(f)≤U(g). Information: g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0 g is discontinuous at every rational number in[0,1]. g is Riemann integrable on [0,1] based on the fact that Suppose h:[a,b]→R is continuous everywhere except at a...

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Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f...

Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f is non-decreasing on [a,b] if and only if f′(x) ≥ 0 for all x ∈ (a,b), while if f is non-increasing on [a,b] if and only if f′(x) ≤ 0 for all x ∈ (a, b). can you please prove this theorem? thank you!

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is...

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is continuous on [a, b]

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