Suppose f and g are contimuously differentiable functions such that f(x) = d(x) and g(x) f(x) and that any product of f, f', g and g' is commutative for all r €R. Show that f - g = C for some real constant C.

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A: Explained below

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A: Solution is given below:

Q: Find the global maximum and minimum of the continuous function f(x) = x²-8 ln x on [1,4]. %3D

A: Given:F(x)=x2-8lnx        1,4

Q: Let ƒ(x), g(x) be two continuously differentiable functions satisfying the relationships ƒ′(x) =…

A: Given,

Q: suppose f is a continuously differentiable function satisfying f(3)=6, and f'(3)=5. Find (f^-1)'(6)

A: Given f(3)=6, f'(3)=5  To find f-1'6 By the formula, f-1'x=1f'f-1x And we know that if f(a)=b then…

Q: 1. Examine the continuity and differential possibilities of function f (x)= |x² – 1| in x= 1.

A: Given To check for continuity at x=1 , we find the limit to the left and right of x=1

Q: 2. Given the function: f(x) 3 x -3 x≤2 2 x² + 1 x > 2 Is f(x) continuous and differentiable at x=2?

A: Here is the solution:

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A: We can show the transitive property of differential function as follows:

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A: Differentiablity analysis problem....

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A: A function f(x) is said to be even if f(-x) = f(x) And odd if f(-x) = -f(x)

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A: Given: The function, y = x-2

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A: We have to find the value of the given integral.

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Q: 2. Determine if a function f : [0, 27] → R defined by f(x) = sin 2n is continuous.

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Q: Suppose that ƒ(x) = x2 and g(x) = | x | . Then thecompositions( ƒ ∘ g)(x) = | x | 2 = x2 and (g ∘…

A: Given: The composite of the two functions are given as,

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Q: Suppose that f is a differentiable function with f(2) = 5 and f'(2) = 7. Let h(x) = f(e*). Find an…

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Q: Show that a function f : R2-R defined by f(x,y)= x3- y3 / x2+ y2, if (x,y) is not equal to (0,0)…

A: What is Continuous Function: A function is referred to be continuous at a certain point in its…

Q: A continuous real-valued function f (x) with f (a) ƒ (b) < 0 , has at least one root in [a, b]

A: This question is related to Real Analysis.

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A: We will show the given statement.

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Q: 3 - x, x<2 Q.No.15: Is f(x) = {+2 x22 continuous at every x € R. x+1 xs1 -.

A: Continuity of a function.

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A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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Q: / Let f: R → R be a differentiable function with the property that f(-x) = f(x) for all x E R. Prove…

A: Let f:ℝ→ℝ be a differentiable function with the property that f-x=fx for all x∈ℝ.   To Prove:…

Q: 1. Suppose that g(x) > 0 for all r, g(x) has a continuous derivative, g(-2) = 5 8 g(4) = 3 and |…

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Q: Use the chain rule to show that for any differentiable function f(x) > 0, f'(x) [In(f(x))]' = f (x)"…

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Q: (b) FIND the value of c, so that f(x) = {cx – 1, x< 0 lx² – 1. x 2 0 is continuous everywhere.

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