Let f, g : [a, b] → R be two Riemann integrable functions such that the set {x ∈ [a, b] : f(x) = g(x)} is dense in [a, b]. Show that the integral f(x) dx = g(x) dx.

Q: Let f be a real valued function defined on [a, b] that is differentiable on (a, b). Suppose f′(x) ≤…

A:

Q: 4. Prove that f(x, y, z) = 2 - 1 x² + y² is continuous in all its domain.

A:

Q: Let F be a continuous function on [a, c} , where a < b< e, and F(z) dz = 6, F(2) dz = 9 A. Evaluate…

A:

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

A:

Q: Prove that the function f: R → R: x x² + ax + b where a, b E R is continuous at every c E R.

A:

Q: Let h : R → R be some twice-differentiable function. Define f(x,y) = h(x) for all x ≠ 0. Show that…

A:

Q: Let f be a real-valued function on [a,b] that is differentiable on (a,b). Suppose f '(x) < 0 for all…

A: This is a problem of the application of differential calculus.

Q: Consider the function f : [−2, 6] → R, x→ f(x) = 2x3/ [√(x2+8)+2]. i) provide argument why this…

A: This is the question of Real topology, and real analysis.

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 […

A:

Q: Let f [0, 6] → R be the function defined as : f(x) = = -2, [4/x], if 0 < x < 1, if 1 ≤ x ≤ 6. Show…

A:

Q: Consider the operator Î = k, where k is a scalar, operating in a function space on the interval [a,…

A: we know that the necessary requirements for L^ linear operators is given by- L^(f(x)+α g(x)=…

Q: 6. Let f, g, h : [a, b] → R satisfy f(x) < g(x) < h(x) for all x E [a, b]. Suppose that f and h are…

A:

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: Show that, if f and g are Reimann integrable on [a, b] then f - g is Reimann integrable on [a, b],…

A: We have to prove that following identity.

Q: Show that a function f : R2-R defined by f(x,y)= x3- y3 / x2+ y2, if (x,y) is not equal to (0,0)…

A: What is Continuous Function: A function is referred to be continuous at a certain point in its…

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…

A: