Problem 5. Let f : R → R be a function, and suppose that f is differentiable at a for anya ∈ R.(a) Prove that for any constant c ∈ R, the function g : R → R, g(x) = f (x) + c isdifferentiable at any a ∈ R with g′(a) = f ′(a).(b) Prove that for any constant c ∈ R, the function g : R → R, g(x) = f (x + c) isdifferentiable at any a ∈ R with g′(a) = f ′(a + c).1
Question
Problem 5. Let f : R → R be a function, and suppose that f is differentiable at a for anya ∈ R.(a) Prove that for any constant c ∈ R, the function g : R → R, g(x) = f (x) + c isdifferentiable at any a ∈ R with g′(a) = f ′(a).(b) Prove that for any constant c ∈ R, the function g : R → R, g(x) = f (x + c) isdifferentiable at any a ∈ R with g′(a) = f ′(a + c).1
Solution
(a) To prove that the function g(x) = f(x) + c is differentiable at any a ∈ R, we need to show that the derivative of g at a, g′(a), exists.
The derivative of g(x) at a point a is given by the limit:
g′(a) = lim (h->0) [g(a+h) - g(a)] / h
Substituting g(x) = f(x) + c into the equation gives:
g′(a) = lim (h->0) [f(a+h) + c - f(a) - c] / h = lim (h->0) [f(a+h) - f(a)] / h = f′(a)
So, the derivative of g at any point a is equal to the derivative of f at that point, which means g is differentiable at any a ∈ R, and g′(a) = f′(a).
(b) To prove that the function g(x) = f(x + c) is differentiable at any a ∈ R, we need to show that the derivative of g at a, g′(a), exists.
The derivative of g(x) at a point a is given by the limit:
g′(a) = lim (h->0) [g(a+h) - g(a)] / h
Substituting g(x) = f(x + c) into the equation gives:
g′(a) = lim (h->0) [f(a+h+c) - f(a+c)] / h = f′(a+c)
So, the derivative of g at any point a is equal to the derivative of f at the point a+c, which means g is differentiable at any a ∈ R, and g′(a) = f′(a + c).
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