F. Let f: R2 → R be a continuous function, and let S be the subset of the R2 given by all pairs (x, y) so that x2 + 3y2 = 12. Show that f(S) contains a minimum and maximum value.

Q: (a) Assume that g(x, y) is a function of the form x²+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that…

A: (a) The given function is gx,y=x3yx2+y2  if x,y≠0,00             if x,y=0,0. To show: (i) gx,y≤xy…

Q: Say if the following statement is true or false: If a function f is twice differentiable at the…

A: If c is a critical point of the function, then  f'c=0. To check whether the critical point…

Q: Find the Wronskian for the set of functions.{x, e−x, ex}

A: Find the Wronskian for the set of functions.

Q: (b) A continuous function f : [0, 1] → R such that its graph (r, f (x)) C R² is not closed.

A: The given statement, "A continuous function f:0,1→R such that its graph x,fx⊂R2 is not closed" is…

Q: x² - y² z = g(x, y) = x² + y² 0 for for (x, y) = (0,0) (x, y) = (0,0)

A: Given function is z=gx,y=x5−y5x4+y4 for x,y≠0,00            for x,y=0,0. We have to check whether…

Q: Let f be a differentiable function of one variable, and let f(p), where p = (x² + y² + z²)'/2. Show…

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Q: Determine the set of points at which the function is continuous. a) f(x,y) = ln (x² + y² – 4)

A: According to question given thatf(x,y)=ln (x2+y2-4)

Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = r? + y (x, y) # (1, –1) x + y is not…

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Q: be real numbers and let f be defined on R by An 5. Let a1, a2, · ...) n f (x) = E(a; – x)², x E R.…

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Q: Let N := {(x, y) e R² | 1 < x² + y? < 4} be a punctured disc and let f : N → R be a continuous…

A: As per the question, we are given a punctured disc (annulus) Ω , on which a locally convex function…

Q: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = %3D (x, y) = (0,0) x²y+2xy² Let…

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Q: f(x)={x^2−9x≤c, 2x−10 x>c } i

A: Consider function f(x)=x2-9           x≤c2x-10        x&gt;c Now  limx→c+ f(x)=limx→c+…

Q: Let g(u) be a differentiable function, and let f(x, y) = g(x^2,y^2)   Find the direction of…

A: Given that gu be the differential function and let fx,y=gx2,y2 We have to find the direction of…

Q: Let h: R → R be a continuous function on all of R. If h(x) T, prove that h(Tt) <0.

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Q: Find the absolute maximum and minimum values of f on the set D. f(x,y) = xy², D = {(x, y)|x > 0, y >…

A: we have to find absolutely minimum maximum

Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = x2 + Y (x, y) # (1, –1) x +y is…

A: Here, the given function is: f(x,y)=1;   (x,y)=(1,-1)x2+yx+y; (x,y)≠(1,-1) lim(x,y)→(1,-1)

Q: 1. Find if the function is continuous in the given points A) f(x, y) = B) f(x, y) = xy √x² + y² 0…

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Q: Show that the function f defined by (x, y) = (1, –1) 22 + y r + y f(x, y) = (x, y) # (1, –1) is not…

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Q: Prove that L(f) ≤ 0 ≤ U(f). Given that the bounded function f : [a, b] → R has the property that…

A: Given that the bounded function f : [a, b] → R has the property that f(x) = 0 for all x ∈ [a, b] ∩…

Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y is…

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Q: Let h: R → R be a continuous function on all of R. If h(x) T , prove that h(T)<0.

A: Given that, let h:ℝ→ℝ be a continuous function on all of ℝ. Also given, hx≤0 for all x&gt;π.

Q: Let f be of the class C" in the interval [a.b] and let c be an interior point of this interval with…

A: We have to Show that if f"(c) &lt;0 then f has a local maximum at x=c. We know that A function f has…

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: Show that the function f defined by (x, y) = (1, –1) f(r, y) = { x2 + y (x, y) # (1,–1) r + y is not…

A: To show that the function f defined by fx, y=1, x, y=1, -1x2+yx+y, x, y≠1, -1is not continuous at 1,…

Q: 2. Determine the points where f is continuous where x + Y if (x, y) # (0,0) f(x, y) = x2 + y2 if (x,…

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Q: Show that the function f defined by (r, y) = (1, –1) (r, y) # (1, –1) f(r, y) = 1² + y is not…

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Q: If m s f(x) S M for a sxs b, where m is the absolute minimum and M is the absolute maximum of f on…

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Q: I. Show that the function f defined by 1, (x, y) = (1, –1) %3D f(r, y) = { r2 + y (x, y) + (1, –1)…

A: Given function is  fx,y=1,             x,y=1,-1x2+yx+y,    x,y≠1,-1

Q: Show that the function f defined by 1, (x, y) = (1,–1) f(r, y) = { x² + y (x, y) # (1,–1) r+ y is…

A: In this question, the concept of limit is applied. Limit Limits are the points at which a function…

Q: Let Y be any topological space and y: [0, 1] → Y be a continuous function so that r(0) = r(1) = xp.…

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Q: Let h: R R be a continuous function on all of R. If h(x) Tt, prove that h(t)<0.

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Q: Consider the function f defined by f(x, y) = x² sin? y. (a) Find and write the critical points of f…

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Q: Show that the function F: C[0, 1] → R defined by F(f) = max f(x) zE[0,1] is continuous.

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Q: Let f be a function in C (R), infinitesimal as x + 0 and with a minimum point at x =0. Then…

A: Given: The function f in C∞ℝ is infinitesimal as x→0 and has a minimum point at x=0   To choose: The…

Q: Let f be a continuous function with f(x) > 0 for all æ € R. Define r(z²+1)" g(r) = e" f(t)dt.

A: Let f be a continuous function with fx&gt;0 ∀x∈ℝ. A function g is defined by  gx=∫0x2+1xexftdt…

Q: Determine the value of m such that the function 5 + 5 y fi^ - ^ - f(z, y) = is continuous at (0,0).

A: Given that function s continuous at (0,0). We need to find the value of m.

Q: For the function f(x, y) = xy((x2 − y2) /( x2 + y2) ) define f(0, 0) such that f is continuous at…

A: A function f is continuous at the origin, the function value at the origin must be equal to the…

Q: 1. Show that if u = x – y, v = y, then for any continuous function f, -" f(r – y, y)dydæ e=s(u+v)…

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Q: 2. Let f: I→R be a continuous function on a bounded, closed interval I= [a, b). Show that f has a…

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Q: Let f be an arbitrary differentiable function of x, y and z, and let x = 2u-v. y =v-2w and z = w-2u.…

A: Given: f is arbitrary differentiable function of x, y, z and x=2u-v, y=v-2w and z=w-2u. To find: The…

Q: Let g(u) be a differentiable function and let f(x, y) = g(x² + y²), find the direction of maximal…

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Q: Determine the set of points at which the function is continuous. F(x, y) = (xy)/(1+e^x-y)

A: A function is said to be discontinuous if it is not defined at a certain point  For the function…

Q: Let f (x, y) = e-(x²+y²), Find the maximum and minimum values of f on the rectangle given below.…

A: We have to find the maximum and minimum value of the given two variables function.

Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y is…

A: Given function is fx,y=1,           x,y=1,−1x2+yx+y  ,x,y≠1,−1 . We have to show that the given…

Q: Suppose that f(x)=xu(x), where u(x) is a differentiable function at x=2 such that u(2)=3, u'(2)=1.…

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Q: Show that the function f defined by 1. (x, y) = (1, –1) f (x, y) : x² + y .2 (x, y) + (1, –1) x + Y…

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Q: Determine the set of points at which the function is continuous. a) f(x,y) = ln (a²² + y² – 4)

A: Given function is                                                              f(x, y)=ln(x2+y2-4)…

Q: There is a number c in the interval [a, b] such that f(c) is the absolute minimum of f on [a, b]

A: Given query is to find statement is true or false.

Q: Assume thatf is differentiable in (-oo, x) and af'(r)N 0 for all a E (-00, 00). Prove that f has the…

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