Question

a) Let f : [a, b] −→ R and g : [a, b] −→ R be differentiable. Then f and g differ by a constant if and only if f ' (x) = g ' (x) for all x ∈ [a, b].

b) For c > 0, prove that the following equation does not have
two solutions. x^{3}− 3x + c = 0, 0 < x < 1

c) Let f : [a, b] → R be a differentiable function and let c ∈ (a, b) such that f ' (c) = 0. If f ' (x) > 0 for x < c and f ' (x) < 0 for x > c then show that f has a maximum value at x = c.

Answer #1

Let a < b, a, b, ∈ R, and let f : [a, b] → R be continuous
such that f is twice differentiable on (a, b), meaning f is
differentiable on (a, b), and f' is also differentiable on (a, b).
Suppose further that there exists c ∈ (a, b) such that f(a) >
f(c) and f(c) < f(b).
prove that there exists x ∈ (a, b) such that f'(x)=0.
then prove there exists z ∈ (a, b) such...

Let f: R -> R and g: R -> R be differentiable, with g(x) ≠
0 for all x. Assume that g(x) f'(x) = f(x) g'(x) for all x. Show
that there is a real number c such that f(x) = cg(x) for all x.
(Hint: Look at f/g.)
Let g: [0, ∞) -> R, with g(x) = x2 for all x ≥ 0. Let L be
the line tangent to the graph of g that passes through the point...

Let A, B ⊆R be intervals. Let f: A →R and g: B →R be
diﬀerentiable and such that f(A) ⊆ B. Recall that, by the Chain
Rule, the composition g◦f: A →R is diﬀerentiable as well, and the
formula
(g◦f)'(x) = g'(f(x))f'(x)
holds for all x ∈ A. Assume now that both f and g are twice
diﬀerentiable.
(a) Prove that the composition g ◦ f is twice diﬀerentiable as
well, and ﬁnd a formula for the second derivative...

Let f : R → R be defined by f(x) = x^3 + 3x, for all x. (i)
Prove that if y > 0, then there is a solution x to the equation
f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove
that the function f : R → R is strictly monotone. (iii) By
(i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why
the derivative of the inverse function,...

Let f, g : [a, b] ---> R continuous such that
(f(a) - g(a)) (f(b) - g(b)) < 0.
a) Show that sup{|f(x) - g(x)| : x ϵ [a, b]} is strictly
positive and
achieved (is a maximum).

Let f : R → R be a bounded differentiable function. Prove that
for all ε > 0 there exists c ∈ R such that |f′(c)| < ε.

Let f : E → R be a differentiable function where E = [a,b] or E
= (−∞,∞), show that if f′(x) not = 0 for all x ∈ E then f is
one-to-one, i.e., there does not exist distinct points x1,x2 ∈ E
such that f(x1) = f(x2). Deduce that f(x) = 0 for at most one
x.

4a). Let g be continuous at x = 0. Show that f(x) = xg(x) is
differentiable at x = 0 and f'(0) = g(0).
4b). Let f : (a,b) to R and p in (a,b). You may assume that f is
differentiable on (a,b) and f ' is continuous at p. Show that f'(p)
> 0 then there is delta > 0, such that f is strictly
increasing on D(p,delta). Conclude that on D(p,delta) the function
f has a differentiable...

Let f, g : [a, b] ---> R continuous such that
(f(a) - g(a)) (f(b) - g(b)) < 0.
b) Show that inf {|f(x) - g(x)| : x ϵ [a,b]} = 0 and is achieved
(is
a minimum).

Let f:[0,1]——>R be define by f(x)= x if x belong to rational
number and 0 if x belong to irrational number and let g(x)=x
(a) prove that for all partitions P of [0,1],we have
U(f,P)=U(g,P).what does mean about U(f) and U(g)?
(b)prove that U(g) greater than or equal 0.25
(c) prove that L(f)=0
(d) what does this tell us about the integrability of f ?

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