Question

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 inverse.

Answer #1

Let f(x) be a continuous, everywhere differentiable function.
What kind information does f'(x) provide regarding f(x)?
Let f(x) be a continuous, everywhere differentiable function.
What kind information does f''(x) provide regarding f(x)?
Let f(x) be a continuous, everywhere differentiable function.
What kind information does f''(x) provide regarding f'(x)?
Let h(x) be a continuous function such that h(a) = m and h'(a) =
0. Is there enough evidence to conclude the point (a, m) must be a
maximum or a minimum?...

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 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 be continuous on [ 0 , ∞ ) and differentiable on ( 0 , ∞ )
. If f ( 0 ) = 0 and | f ′ ( x ) | ≤ | f ( x ) | for all x > 0 ,
then f ( x ) = 0 for all x ≥ 0 .

Let f: [a,b] to R be continuous and strictly increasing on
(a,b). Show that f is strictly increasing on [a,b].

Let f: R --> R be a differentiable function such that f' is
bounded. Show that f is uniformly continuous.

. Let f and g : [0, 1] → R be continuous, and assume f(x) = g(x)
for all x < 1. Does this imply that f(1) = g(1)? Provide a proof
or a counterexample.

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. x3− 3x + c = 0, 0 < x < 1
c) Let f : [a, b] → R be a differentiable function...

Let g from R to R is a
differentiable function, g(0)=1, g’(x)>=g(x) for all x>0 and
g’(x)=<g(x) for all x<0. Proof that g(x)>=exp(x) for all x
belong to R.

Consider the function f : R → R defined by f(x) = ( 5 + sin x if
x < 0, x + cos x + 4 if x ≥ 0. Show that the function f is
differentiable for all x ∈ R. Compute the derivative f' . Show that
f ' is continuous at x = 0. Show that f ' is not differentiable at
x = 0. (In this question you may assume that all polynomial and
trigonometric...

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