Suppose f is differentiable at a and f(a) + 0. (a) Prove |f| is differentiable at a by using that |f| = /f². What are the possible values of |f|'(a)? (b) Give a counterexample that shows the conclusion need not hold if f(a) = 0. Identify where your calculations for (a) fail in this case.

Q: 1. Express the followings in the form of x + iy where r, y E R.(choose one.) (a) 금+ 부 (b) ( + ) ( –…

A: Rationalise  all the term and calculate to get the final answer.   Actually as per the barteleby…

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A: First we find out f'(x) then put x=a=3 in f'(x).

Q: If f is concave up, then f' is (choose one): (a) increasing. (b) decreasing.

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A: Note: As per our guidelines we are supposed to answer only 3 subparts if there are multiple subpart…

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Q: 1. Sketch a function f(x) with the following properties: f'(x) > 0 and f"(x) = 0, for r 2 . .

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Q: Please just do 1(b)

A: To compute the required value , by definite integral

Q: 11. Prove from the definition that the following functions are differentiable. ()- (b) f =xy?

A: By the definition, the function f(x,y)=xy^2 is differentiable if fx(x,y) and fy(x,y) both exist.

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Q: Let f(x) = 2 – 2², x > 0 f'(x) = Preview

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Q: 7. Show that if f is a function of r only, then 0²ƒ(r) = f'(r) + ²ƒ'(r)

A: The given data is: Show that if f is a function of r only, then ∇2fr=f''r+2rf'r