12. If a function f is continuous in [a, b] and c e ]a, b[ such that f (c) # 0, then there exists some 8> 0' such that f (x) has the same sign as flc) for all rs lc -8 c+ 8r

Q: 1. Supposef is a continuous function on x 2 0,f'exists for x> 0, f(0) = 0 and f'is monotonically…

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Q: The following statement is false, explain why. The function g(x) = x^2 on the interval [-1,1] has…

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Q: 1. Show that for all f e L'(R) lim dr = 0. %3D

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Q: (5) Use the limit definition of derivative to prove the following: (a) For any a e R, (f(ax))' =…

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Q: '3 Suppose that f" is continuous on [1, 3] and f(1) = 8, f(3) = -3, f'(1) = 7, and f'(3) xf "(x) dx.…

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Q: 1. Let the function f: R R be given by 1- cos(x) if x + 0, f(x) := if x = 0. 1 Prove that f is…

A: Differentiablity analysis problem....

Q: 16. Suppose that f is differentiable at c and that f'(c) + 0. Show that there exists a ð > 0 such…

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Q: Suppose that ƒ is continuous in [0,1] and that f(x) E [0, 1] for all x. Show that f(x) = x for some…

A: Given that f:0,1→0,1 is continuous on 0,1. To show fx=x for some x∈0,1

Q: Suppose that the function f: [0, 1] in R is continuous, f(O) > 0, and f(l) = 0.Prove that there…

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Q: If f(x) is a continuous function on [3,7] and F(x) is an anti-derivative of f(x) such that F(3)=7…

A: Given :f(x) is continuous function 3,7Fx is anti-derivative of fxF3=7 and F7=3To determine…

Q: 2. Let f(x) = V4 – x for x < 4 and g(x) = x² for all x E R. (a) Give the domain of f · g, ƒ + g, ƒ º…

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Q: Suppose that a function ƒ is continuous on the closed interval [0, 1] and that 0&lt;=ƒ(x)&lt;= 1 for…

A: Let us assume a function g(x)=f(x)-x Since both f and x are continuous on the closed interval [0, 1]…

Q: 2. Consider the function if x = in lowest terms q > 0, p, q € Q; f(x) = O if x e Q°,x = 0 Question:…

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Q: 45. For what value of a is x2 - 1, x < 3 2ax, f(x) = %3D 2ах, continuous at every x?

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Q: 1. Suppose that f : [0, 0) → R is continuous and f(x) # 0 for all x > 0. If we have (f(x))? = 2…

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Q: Suppose f : R → R is a continuous function with the property that f(x) = 0 for every x E Q. Prove…

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Q: 4. Find the maximum value of f (x) = In(x – In x) on the interval -sse. e

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Q: 2: If f(x) and g(x) are continuous on (a,b]and if f =[s . Prove that 3cela,b] such that f(c)=g(c).…

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Q: 19.D. Show that the function defined by g(x) g² sin (1/x), x * 0, %3D = 0, 0, is differentiable for…

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Q: Can Rolle's Theorem be applied to f(x) = 1/(1-x) on interval [0,3]? If yes, what is c in (0,3) such…

A: here we can't apply the rolles Theorem

Q: 9. Consider the function f: R → R, defined by (x²sin(1/x) ifx # 0 f(x) = if x = 0 Then a. f is not…

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Q: Assume that ƒ is continuous on [0,1] and that f(0) = 2 and f(1) = 1. Show that the graph of y = f(x)…

A: given  f is continuous on [0,1] f0=2,f1=1y=x2+1

Q: Find a value of k making h(x) continuous on [0, 5]. kx 0 < x < 2 x + 1, 2<x < 5 h(x) =

A: The given function is hx=ekxif 0≤x&lt;2x+1if 2≤x≤5. The function is continuums. That is,…

Q: Consider the function f(x) = 5 – 5x2/3 on the interval [-1, 1]. Which of the three hypotheses of…

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Q: Suppose that f:[0, 1] → R is continuous on [0, 1] and differentiable on (0, 1) with f (0) = 0 and…

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Q: 35. Given that f(z) is a continuous even function, then A 0 D

A: We will use the definition of an even function and properties of definite integrals to find the…

Q: 4. For a function f:D → R and a point xo in D, define A = {x in D x ≥ xo} and B: {x in D x ≤ xo).…

A: Recall definition of conituity as, Right hand limit = left hand limit = value of function at the…

Q: 7. Suppose f is a continuous function on [a, b] and Sf(x)g(x) dx =0 for every continuous function g.…

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Q: 29. Prove that f(x) =x² is uniformly continuous in 0 0, we can find 8 > 0 such that |x – x <e when…

A: to show f is uniform continuous on a set A , we need to show that for any given epsilon positive…

Q: 4. Suppose that f is continuous on a closed interval [a, b] and differentiable on the open interval…

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Q: Suppose f is known to be differentiable and contains a local extremum at c, then which of the…

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Q: 1. Determine whether k(x) is continuous at a = 3, where x + 3, x 2.

A: Left-hand limit of f(x) at x=a is represented by limx→a- f(x) and it can also written as limh→0+…

Q: 5.9. Suppose that f is continuous at zo and g is continuous at wo = ). Prove that the composite…

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Q: Suppose that f (0) = 2 and f' (x) ≤ 3 for x &gt; 0. Apply the MVT to the interval [0, 4] to prove…

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Q: If f(x) is a continuous function on [4,7] and F(x) is an anti-derivative of f(x) such that F(4)=9…

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Q: Show that the converses of parts (a) and (b) of Theorem 6.2.8 are false by finding counterexamples.…

A: Theorem: Let f be a differentiable function on the interval I. Then, If f'x&gt;0 for all x∈I, then…

Q: 3. Use the MVT to show that the function f(x) = 1/x³ is uniformly continuous on the interval 3,00).…

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Q: - 35) Prove that if y=flx)is continuoUS and flx1= 0 when x is a rational number, then f Lx)=0 for…

A: The objective is to show f(x) =0 for all x belongs to R.

Q: 2. Consider the function f with domain (-2, 0): 3|22 – 1| if – 2<r<1 2.r2 + 2x – 4 f(x) = 3 if r = 1…

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Q: Show (in terms of  −δ) that a function f : [2,7] −→R be defined by f(x) = √x2+1 is uniformly…

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Q: 13. Let g be defined by V7x +2 V6x + 4 | g(x) = if x 2 - and x 2 x – 2 k , if x = 2 For what value…

A: The given functiong(x)=7x+2-6x+4x-2,if x≥-27 and x≠2k,if x=2

Q: 6.2.5 Theorem Suppose that f is continuous on the closed interval I := [a, b), that f is…

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Q: II. Prove the following theorems on limits. 1. f(x) = L if and only if (x) – L] = 0 2. f(x) = L if…

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Q: 9. Consider the function f(x, y) = p |xy|. Show that Taylor’s expansion about the point (x, x) is…

A: The Taylor series expansion of a function f(x ,y) about point (a ,b) is given by,…

Q: Let f be a differentiable function with f(0) = 1 and / f'(1) dt = x². Then Of(x) = COS X Of(x) = (x+…

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Q: (1- a? f(x) = 1– x² if x E Q, if r ¢ Q. %3D p at which f is continuous. Prove your conclusion.

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Q: For f(x) = √ √2-4 4- ™ Of is not continuous at x = 4 Of is continuous at a Of is continuous at a 5…

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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…

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Q: 3. Consider the function g(x) defined by { x² sin(1/x) if x > 0, g(x) = if x <0. (a) Show that g(x)…

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Q: 2. Consider the function f(x) = if x = {9 in lowest terms q > 0, p,q e Q; 0 if x e Q°,x = 0…

A: We use sequential approach of continuity to show it.