Question

Consider the piecewise defined function

f(x) = xa− xb if 0<x<1. and

f(x) = lnxc if x≥1.

where a, b, c are positive numbers chosen in such a way that f(x) is differentiable for all 0<x<∞. What can be said about a, b, and c?

Answer #1

Consider the function f defined on R by f(x) = ?0 if x ≤ 0, f(x)
= e^(−1/x^2) if x > 0.
Prove that f is indefinitely differentiable on R, and that
f(n)(0) = 0 for all n ≥ 1. Conclude that f does not have a
converging power series expansion En=0 to ∞[an*x^n] for x near the
origin. [Note: This problem illustrates an enormous difference
between the notions of real-differentiability and
complex-differentiability.]

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...

If f is a continuous, positive function defined on the interval
(0, 1] such that limx→0+ = ∞ we have seen how to make sense of the
area of the infinite region bounded by the graph of f, the x-axis
and the vertical lines x = 0 and x = 1 with the definition of the
improper integral.
Consider the function f(x) = x sin(1/x) defined on (0, 1] and
note that f is not defined at 0.
• Would...

fourier expansion, piecewise function.
f(x){ pi , -1<x<0
-pi , 0<x<1

Let f(x) be a twice differentiable function (i.e. its first and
second derivatives exist at all points). (a) What can you say about
f(x) when f 0 (x) is positive? How about when f 0 (x) is negative?
(b) What can you say about f 0 (x) when f 00(x) is positive? How
about when f 00(x) is negative? (c) What can you say about f(x)
when f 00(x) is positive? How about when f 00(x) is negative? (d)
Let...

Let a>0 & b>0 be two positive numbers and consider the
function f(x) = x^a+x^−b. Find the positive value of x where f(x)
achieves its minimum value.
a. x=1
b. x=a/b
c. x=(b/a)^1/a+b
d. x=(ab)^a+b
e. x=(a+b)^ab
f. x=(b/a)^a+b

The function f(x) = 3x 4 − 4x 3 + 12 is defined for all real
numbers. Where is the function f(x) decreasing?
(a) (1,∞) (b) (−∞, 1) (c) (0, 1) (d) Everywhere (e) Nowhere

If f is a differentiable, real-valued function such that f′(c)
< 0 and f′′(c) < 0, what can be said of f at the point c?

3. For each of the piecewise-defined functions f, (i) determine
whether f is 1-1; (ii) determine whether f is onto. Prove your
answers.
(a) f : R → R by f(x) = x^2 if x ≥ 0, 2x if x < 0.
(b) f : Z → Z by f(n) = n + 1 if n is even, 2n if n is odd.

Let f : [0,∞) → [0,∞) be defined by, f(x) := √ x for all x ∈
[0,∞), g : [0,∞) → R be defined by, g(x) := √ x for all x ∈ [0,∞)
and h : [0,∞) → [0,∞) be defined by h(x) := x 2 for each x ∈ [0,∞).
For each of the following (i) state whether the function is defined
- if it is then; (ii) state its domain; (iii) state its codomain;
(iv) state...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 5 minutes ago

asked 11 minutes ago

asked 22 minutes ago

asked 22 minutes ago

asked 31 minutes ago

asked 32 minutes ago

asked 50 minutes ago

asked 50 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago