: Assume that a bounded function f [a, b] → R is integrable. If f(x) >0 for all ze [a, b], then prove that f(x)da ≥ 0.

Q: (a) Write the equation in slope-intercept form, and determine the slope and y-intercept. (b) Graph…

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Q: Suppose f is integrable on [-b, b] and f is an odd function-that is, f(-t) = -f(t) for all tE(-b,…

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Q: Let f : [0, 1] → R be a continuous function. Show that the function g : [0, 1] → R given by g(x) =…

A: The given function: is continuous. That is, assume f(x)=x, then x is Real number. Show that the…

Q: Let f: X→Y be a function, and let A,BCY. Prove or disprove: f-¹(A)uf-¹(B).

A: The given statement : f-1(A ∪ B) = f-1(A) ∪ f-1(B) is true and we shall prove this statement.

Q: Consider the function f : [0,3] –→ R defined by f (x) = 1 for x E [0, 1), ƒ(x) = 2 for x E [1,2),…

A: We are given a function f:[0,3]→ℝ defined as,   f(x)=1 if 0≤x<12 if 1≤x<23 if 2≤x≤3 We need to…

Q: Indicate whether the following statement is true or false. Let f(x, y) be any c' real valued…

A: Given fx,y be the C1 real valued function defined on the rectangle R=a,b×c,d. Then we have to check…

Q: Let f be a real valued function defined on [a, b] . Let f have a local minimum at a point x ∈ (a, b)…

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Q: Let f : [a, b] → R be increasing on the set [a, b] (i.e., f(x) ≤ f(y) whenever x < y). Show that…

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Q: Let f : R² → R² be a continuous function. Prove that f([0, 1] × [0, 1]) is compact.

A: It is given that f:ℝ2→ℝ2 be a continuous function. Prove that f0, 1×0, 1 is a compact set. Let Gi:…

Q: B). For what values of r, reR is the function [(x+y+z) ƒ(x,y,z) = { x² + y² +z² 0, continuous on R.…

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Q: 1. Suppose f: [a, b].→ R is C¹ and that f(a) = f(b) = 0. Show that 2 (S f(x)dx) a 6³ - a³ 3 rb […

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Q: Show that at (0,0). f(x, y) = x² sin 0, (²7)(y² +1), (x, y) ‡ (0,0), (x, y) = (0,0). of (0,0)…

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Q: Assume that f : R → R is differentiable everywhere and satisfies for all h, x E R: f(x + h) – f(x) =…

A: Let's prove this.

Q: Let R = [a, b] × [c, d] be a rectangle. Also suppose that f (x) is continuous on [a, b] and that…

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Q: Let h : C N x N where C CZ given by h(m) = (m2 +1, m2 + 2). Choose all subsets C' such that h is an…

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Q: Prove that for any Riemann integrable function f on the interval [a, b] there is c such that f = [°…

A: let f be Riemann integral an [a, b] then clearly f is bounded.

Q: Let f, g, h be functions on [a, b] such that f(x) ≤ g(x) ≤ h(r). Prove that if f and h are…

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