Let f : (a, b) → R be continuously differentiable, where −∞ ≤ a < b ≤ +∞. Consider x0 ∈ (a, b) Show that f is locally Lipschitz around x0 i.e. there is a δ > 0 and a C > 0 such that |f(x) − f(y)| ≤ C|x − y|, ∀x, y ∈ [x0 − δ, x0 + δ] Does this imply that f is Lipschitz continuous throughout (a, b)? If no, provide a counterexample.

Q: Let f € L'(0,1). Assume that f(t)t2"dt = 0 for all n f = 0 a.e. 1,2,3,.... Prove that %3D

A:

Q: Suppose f'(x0) = 0, f"(xo) = 0, ., f(n-1)(xo) = 0 and f(") (x0) > 0, for a Cn function f. Prove that…

A:

Q: Suppose f and g are continuous functions on [a, b] and that g(x) > 0 for all x E [a, b]. Prove that…

A:

Q: Let $c$ be a critical number of a function $f$ that is continuousinterval, except possibly at $c$,…

A:

Q: If f(x) = sin(x), g(x) = ex, then the wronskian of f and g at point x=0 is W[f,g](0)

A: The wronskian of f and g is given by Wf, g = f(x)g(x)f'(x)g'(x)

Q: Let P(z) = | tin(t) dt. Use part 1 of the fundamental theorem of calculus to determine if P(z) is…

A:

Q: Let f(x, y,z) = x² sin y + y e², where x =t + 3,y = net-1 and z = In 2t. dL at t = 1 is: 1-2n O -12n…

A: Given, fx,y,z=x2siny+yez, where x=t+3,y=πet-1 and z=ln 2t.   The objective is to determine dfdt at…

Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

Q: Let BCR be the open ball of radius one centered at the origin, i.e. B = {(x₁,...,xn) € R" : x² + +…

A: As per the question we are given a n dimentional open ball B of radius 1 centered at the origin. And…

Q: Let a be a conostant and let f(x)={x^(3)-ax, if x=1 then (a) f is continuous at 0 only if a=0 (b)…

A:

Q: Observe that f (x) = x 2 is continuous on [0, 1] with f(0) = 0 and f(l) = 1. Because f (0) &lt; 0.5…

A: f(x)=x2 is continuous on [0,1] with f(0) = 0 and f(l) = 1 we need to prove that according to…

Q: Consider the Cauchy Problem y 0 = a(x) arctan y, y(0) = 1, where a(x) is a continuous function…

A: We have the given Cauchy problem is y'=a(x)tan-1y,  y(0)=1   …..(1) where a(x) is a continuous…

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

A:

Q: Let A(x) = S f(t) dt, with f(x) as in figure. 1. 1. 0,5 -1 -2 A(x) has a local minimum on (0, 6) at…

A:

Q: Consider the following function: f(x,y) = { (x2+y2)sin[1/(x2+y2)], for (x2+y2) not equal to 0; 0,…

A:

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

A:

Q: Let A(x) = S° f(t) dt, with f(x) as in figure. %3D 1,8 - 0,5 -1 A(x) has a local minimum on (0, 6)…

A: Given: A(x)=∫0x f(x)

Q: Define f : X → K by f(x) = x'(1), x € X. Again, the linear functional f is not continuous. By 6.4,…

A: Given: f:X→K by fx=x'1, x∈X To prove f is not continuous using the given corollary.

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: Given F(x, y) = (e*y°(cos(x) + ¢(x)), 2ye*o(x)), with h E C'(R). then O (A) it does not exist o such…

A: According to our QnA guidelines, I can solve only one question .Please repost other question.

Q: Suppose that |- f(x) is not constant. Il- f(x) is differentiable on 0, o0 II- S f(t)dt = (x +1)f(x)…

A: In the given question we have to choose a function which satisfies the given three condition in the…

Q: Let F(x, y, z) = z* +x+y sin x +x In z 0, where z =f(x, y), then at (0,0,1) and %3D are equal to:

A: Given,          Fx,y,z=z4+x+ysinx+xlnz=0 , where z=fx,y

Q: If f is continuous on the closed interval [a, b], which of the following must be true: (a) f has at…

A:

Q: At what points (x,y,z) in space are the functions continuous? a. h(x,y.z) = In (3z° - 4x- 5y- 4) 1…

A: Please rate

Q: If R is a closed region on the xy-plane andf is a continuous function on R, then f has a relative…

A: The given statement is True.

Q: Find the Wronskian for the set of functions.{x, ex, sin x, cos x}

A:

Q: Show that the Dirichlet function f defined on [0 , 1] by f(x) = { 1 if x is rational 0 if x is…

A:

Q: Q2: If f(x) is Riemann-Stieltjes integrable function with respect to g(x) on [a,b] and if…

A: It is given that,  f(x) is Riemann-Stieltjes integrable function with respect to g(x) on a, b such…

Q: Let f be a continuous (and therefore uniformly continuous) function on [a,b). Show that, given e> 0,…

A: Consider the given information.

Q: Prove that the function f(z2) defined by x*(1+ i) – y°(1 – 1) , (z ± 0) x² + y? f(z) = [0, (z = 0)…

A:

Q: Let F be defined on R x R to R by F(x, y) = x² – y. Show that F is C' on a neighborhood of (0, 0)…

A: Let F be defined on RXR→R by F(x,y) =x2-y. Now we have to show that F is in class C' on a…

Q: Let f(x, y) = x² + y² – 2a°y- if (x, y) # (0,0), - (14 + y²)² ' and f(0, 0) = 0. Show that f is…

A: Given: fx,y=x2+y2-2x2y-4x6y2x4+y22 if x,y≠0,0 and f0,0=0.   To show: (a.) f is continuous at the…

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

A:

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 (x) be a continuous function on R-{0} such that a a f (x) dx exists for all a > 0.If g(x) = [?…

A:

Q: (a) Let Ty if +y +0, g(x, y) = 0 if r = y = 0. Prove that g(r, y) is continuous at (0,0).

A: To Prove: g(x,y) is continuous at (0,0)

Q: Let F(x,y,z) = z* + x+ y sin x + x In z 0, where z fx.y), then at (0,0,1) and are dy equal to: ax

A:

Q: 3. Prove by using the Squeeze Theorem, or otherwise: 1 t2 dt is continuous at 0. T sin t (a) (b) 1+…

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: Consider the function { that is defined on R as: f(x)=-x² +4x-3. Designate by (C) its representative…

A: The given function is, fx=-x2+4x-3 The orthonormal system is, O; i→,j→ To evaluate limx→-∞fx  and…

Q: For a function h and an interval I, the oscillation of h on I is defined by w(h, I) = sup |h(æ) –…

A:

Q: Let A(x) = So f(t) dt, with f(x) as in figure. 1.0 0.5 A(x) has a local minimum on (0, 6) at x =…

A:

Q: (3x² -1, x 1, Determine c and d so that f(x) is continuous everywhere.

A:

Q: A:= {(x, y) € R² :1< log(x) < 2}. = {(x, y, z) E R³: sin(xy² + «/2) · cos(z) = 1} . (a) Decide…

A:

Q: Let n e N, let U C R" be open and let f : U → R be partially differentiable. Assume that there…

A:

Q: Prove this Let E be an open subset of R" , and let f : E → R™ be a twice continuously differentiable…

A:

Q: If f:[a, b] →R has a local minimum at p E (a, b) and f is differentiable at p, then prove that f'(p)…

A:

Q: Q2: If f(x) is Riemann-Stieltjes integrable function with respect to g(x) on [a,b]and if…

A: detailed explanation mentioned below.

Q: Let f(x) be continuous on [0, oo) and twice differentiable on (0, co). If f'(x) > 0 and f"(x) 1,…

A: Let f(x) be a continuous function on [0,∞) and differentiable at 0,∞. The given double derivative…

Q: 15. Let f. g : R → R be continuous at a point c, and let h(x) := sup (f (x). g(x)} for x e R. Show…

A: