If f is analytic in a neighborhood of ro and f(xo) = 0, show that f(r)/(x – ro) is analytic in the same neighborhood.

Q: Determine the domain of the function and prove that it is continuous on its domain using Theorems…

A: Given: f(x) = ln (9 - x 2) To determine: The domain of the function and prove that it is continuous…

Q: Let f (x) be a function satisfyingIf (x)| <x? for -1<x< 1. Show that f is differentiable at x = 0…

A: The solution is in step 2

Q: Find the antiderivative F of f(x)=4-3(1+x^2)^-1 that satisfies F(1)=-2.

A:

Q: Show that the function f(x) = e¯*/2 has inflection points x = f1. at

A: We have the given function as fx=e-x22              ⋯⋯1 As we know that, a point of inflection of…

Q: Is there a real function f(x) that is differentiable only at a point x?

A:

Q: b) Suppose that f(x) 1 if x is negative. Can ƒ be continuous at x = 0?

A:

Q: If f is continuous and ∫31 f (x) dx = 8 , show that f takes on the value 4 at least once on the…

A: Given that f is continuous and ∫31 f (x) dx = 8. We have to prove that, f takes on the value 4 at…

Q: If f is analytic in a neighborhood of xo and f(xo) = 0, show that f(x)/(x – xo) is analytic in the…

A:

Q: If f is differentiable at a number, x, then f is continuous at x.

A:

Q: if f(x) is a function that is continous at (2,f(x)) then f(x) is differentiable at (2,F(2))   true…

A: Given statement is If f(x) is a function that is continous at (2,f(x)) then f(x) is differentiable…

Q: Find the value a that will make f(x)={2x^2-7x+3/x-3  if x is not equal to 3…

A:

Q: A. Consider the function defined by [x² +1, if x<5 f (x)={ |6x– 4, if x25 a. Is f continuous at x =…

A:

Q: Suppose that f is a function on the interval [1, 3] such that –1 < f(x) < 1 for all x and f(x) dx…

A: It is given that ∫13fxdx=0 and -1≤fx≤1.

Q: Let f be a differentiable function such that f(4) = 3 and f'(x) = -3x + 3. What is the approximation…

A:

Q: Show that f is not differentiable at x = 1 and has a corner in its graph there. x 1

A:

Q: Let 'f be a continuously differentiable real valued function defined bx + a by f(x) ={ 5x x<1 x21…

A:

Q: Determine the domain of the function and prove that it is continuous on its domain f(x) = e - x 2

A: Consider the given function. fx=e-x2 The function is define for every real value of x. Hence,…

Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: 2) Consider the piecewise function: |x – 1| x² – 1' x 1 For what real number A is the function…

A: A function f(x) is continuous at x=a if the left-hand limit and the right-hand limit of the function…

Q: (a) Let x2 if x is rational f (x) = 0 otherwise. Prove that f is differentiable at 0.

A:

Q: Show analytically that the function f is not differentiable at x = 0. Ax) = 2x+1 x>1

A:

Q: Consider a function f(x) = |x – 3|. a. Show that f (x) is not differentiable at 3. -

A: Given         f(x) =|x-3|

Q: Find the best quadratic approximation to f on the interval [ 0, 1].                     f(x) = x3

A:

Q: If f is differentiable at xo, prove that f (xo + h) – f (xo – h) – f' (x0) - lim h-0 2h

A: Given that: f is differentiable at xo   To prove: limh→0f(x0+h)-f(x0-h)2h=f'(x0)

Q: Given f(x) = |2x – 2|. 1) At which points is f not differentiable? Prove your claim using the limit…

A:

Q: Consider the function (x)=x²+2x . Using the limit definition of the derivative, show that for any…

A:

Q: Show that the function is differentiable at x=0 and find f'(0). f(x) = x2sin(1/x), if x is not…

A:

Q: If f(2) is a continuous function at a point z, , then f(z) is differentiable at z, . But the…

A: Authoring guidelines:…

Q: Suppose f and g are differentiable functions on R such that, (x²+1)f (x) + 3xf(x) = e*2 + 7x and…

A:

Q: show that f,given by f(x) x +0 )is continuous on R-{0}

A:

Q: Let f(x)={tan(2x) ax+b if x>0 if x<0 For which values of a and b is f(x) differentiable for every…

A:

Q: Let f be a differertiable function such that f(-5) = -9 and f'(x) = 2x + 8. What is the…

A:

Q: Show that the function f(x) = 1 – I - x² is continuous on the interval [-1, 1].

A:

Q: Use the intermediate value theorem to show that f(x) = (e^−x^2)−x has at least one root in the…

A:

Q: Let be a differentiable function such that f(3)=2.345 and f'(x)= ln(x^2+1).What is the value of…

A:

Q: Use the intermediate value theorem to show that f(x) = e −x 2 − x has at least one root in the…

A:

Q: Find the constant a such that the function is continuous on the entire real line. 5x², ах — 7, f(x)…

A: For a function to be continuous, the value of the left and right limits at critical point needs to…

Q: Let [x² + 1 _ if x 1 f(x) - Is f differentiable at 1?:

A:

Q: Is the function f(z) = (x2 - y2) + 2xyi entire? Why or why not?

A:

Q: Find the values of a and b such that the following function f(x) = 2a cos(x + 1) - b.5x+2, if x -1,

A:

Q: Find the constant a, such that the function is continuous on the entire real line. ( 2x², f(x) = if…

A: Given the function is f(x) = 2x2,if  x≥1ax-1,if  x&lt;1

Q: Consider the functions f(x)= x3-1 and g(x)=−2x+sin(x). Show that there is exactly one C in (−∞,∞)…

A:

Q: Find the values of a and b such that f is differentiable at x = 3

A: Given,            The function is fx=ax+bx≤32x1-1x&gt;3To find : Find the values of a and b such…

Q: Use the e-d definition to prove that f(x) = x³ + 3 is continuous at xo = 3.

A:

Q: .For what values of a is Jaax - 2a, x z 2 f(x) 12, x < 2 continuous at every x?

A: Here given function  Since,  it is continuous for every x thus it Is also continuous for x=2