(a) Let g : [0, a] → R be differentiable, g(0) = 0, and|g'(x)| ≤ M for all x ∈ [0, a]. Show  |g(x)| ≤ Mx for all x ∈ [0, a].

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Q: 7. A function f : (a,6) → R satisfies a Lipschitz condition at x € (a,6) iff there is M > 0 and e >…

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Q: snip

A: The given statement is true. Because the local maximum is the highest point on the curve. so that…

Q: Discuss limit(n →∞)fn on B and C for  fn(x)=x/n; B=E1; C = [a, b] ⊂ E1 fn(x)=(cos x + nx)/n; B=E1

A: Note: Since you have not mention anything about the set E1, We are considering it any sunset of real…

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A: Given equation is Differentiating both sides, we get Now, we will wind critical points of f(t) in…

Q: 2. cuts the curve at Q, such that Let ƒ be a continuous and differentiable Tunction in (x,, x.). If…

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Q: 4- If a function is differentiable at zo, then it is : a) Differentiable on C b) Continuous on C c)…

A: To answer the given question about the complex function.

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A: I give two examples for this statement.

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Q: Let k(x) = h(x) - g(x), where g and h are infinitely differentiable functions from R to R. Define…

A: Let kx=hx-gx, where g and h are infinitely differentiable functions from ℝ→ℝ. Let fn be the nth…

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Q: 6. Let 1 f (x) - cos x + sin(2x), x E [0, 1] 2 Find the least squares polynomial approximation of…

A: Let fx=12cos x-sin2x, x∈0,1. To find quadratic fitting  fx≈P2x=a0+a1x+a2x2 Consider, s0=∫011dx=1

Q: (a)differentiable at x = 0, if a = 0 and b = 1 (b)differentiable at x = 1, if a = 1 and b = 0…

A: Hello! Thank you for the question but as per our honor code, we'll able to solve only the first…

Q: 3.x2 Let g(x) = 9. x > 0 -2x2 x <0 (a) Prove that g is differentiable on R.

A: A f(x) function f is differentiable at x iflim h→0 f(x+h)−f(x)h exists. i.e. if lim h→0+…

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Q: [0, 4). 4. Prove that f(x) = x² is both uniformly continuous and continuous on

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Q: Let f(x) be differentiable on R. Suppose f(0) = A > 0 and lim f(x) = lim f(x)=0 1. Show that f(x)…

A: Solution: Theorem (Rolle's Theorem): Let a function f:a,b→ℝ be such that (i) f is continuous on a,b,…

Q: please answer thank you!

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Q: (b) A function f(x) continuous at 0 and g(x) not continuous at 0 such that f(x)+g(x) is continuous…

A: A function fx is continuous at  x=a, if limx→a+f(x)  = limx→a-f(x) = f(a)

Q: 9. If f(x) is a continuous function on [a, b], differentiable function on (a, b), then there is at…

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Q: 3. Let f : R→ R be continuous and let a > 0. Define g : R → R by rx+a g(x) = f(t)dt, x € R. x-a Show…

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Q: 8. Let f,g be differentiable on R and suppose that f(0) = g(0) and f'(x) 0. Show that f(x) 0.

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Q: 1. Determine whether the functions f1(x) = cos 3x, f2(x) = x, and f3(x) = cos² x are linearly…

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Q: a. We say that a function ƒ : [a,b] that f(x) 0 > Df(x). R has a local minimum at x if there exist…

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Q: V4 - a2 Given f(x) and g(x) = |x|, fogis continuous on the interval %3D 이-2,-1] U [1,2] o[-2, 1) U…

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Q: f(2) = e=².

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Q: 2. Suppose that f is a twice differentiable function (everywhere). The so-called arclength kernel of…

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Q: 7. Let f(x, y) = In(x)+2 ln(y) – x – 4y. Find the critical point of f and determine if it is a local…

A: Let's find.

Q: 4. If continuous function f:[0,1] U (2,3) → R is one-to-one, is f also continuous? 1

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Q: (c) Suppose that f : [-1,1] →R is a continuous function such that r²+ f(x)² = 1 for every rE[-1, 1].…

A: Given that the domain of the given function is [-1, 1] and function is defined as: f:−1,1→ℝ​is a…

Q: 2. A function on the interval [0, 1] is Lipschitz if it satisfies the condition If(s) – f(t)| <C\s -…

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Q: rove for any x > 0, x/(1+x) < ln(1+x) < x

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Q: (c) A function f(x) continuous at 0 and g(x) not continuous at 0 such that f(x)g(x) is continuous at…

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Q: 2. Given g '(x) : 8 – 2x and g is continuous on its domain of (0,0), find the critical values of g.…

A: The objective is to find the critical values of g.

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

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Q: 1-x², x 2 at x = 2.

A: Given, fx=1-12x2, x&lt;23,    x=2x-2-1,   x&gt;2

Q: J arctan x dx A C In(x2) + C In(x² + 1) + C x arctanx x arctan X - | 1. x arctan x +- In(x2) + C x…

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Q: 9. If f(x) is a continuous function on [a, b], differentiable function on (a, b), then there is at…

A: Given that: f(x) is continuous function on [a,b], differentiable function on (a,b),

Q: If f(x) is continuous over an interval [a,b], and the function F(x) is defined by F(x) = [ f(t) dt,…

A: This is a problem related to the trigonometric integral calculus. Based on the general fundamental…

Q: f(x) = 3 x4-4 x3-12 x2+1, has local minimum at x= 1 40 2.80 .CO .DO 0.

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