Question

(a) Show that the function f(x)=x^x is increasing on (e^(-1), infinity)

(b) Let f(x) be as in part (a). If g is the inverse function to f, i.e. f and g satisfy the relation x=g(y) if y=f(x). Find the limit lim(y-->infinity) {g(y)ln(ln(y))} / ln(y). (Hint : L'Hopital's rule)

Answer #1

Show that f: [1, infinity) --> [2, infinity), f(x) = x+1/x is
bijective and find it's inverse function.

1. Consider the function f(x) = x 3 + 7x + 2. (a) (10 pts) Show
that f has an inverse. (Hint: Is f increasing?) (b) (10 pts)
Determine the slope of the tangent line to f −1 at (10, 1). (Hint:
The derivative of the inverse function can be found using the chain
rule.)

Show the following:
a) Let there be Y with the cumulative distribution function
F(y). Let F(Y)=Z. Show that Z~U(0,1) for F(y).
b) Let X~U(0,1), and let Y := -ln(X). Show that Y~exp(1)

the function g(x) is increasing on (-infinity, 2) and (5,
infinity). The function is decreasing on (2,4) and (4,5). The
function is concave down on (-infinity, 3) and (4,5) and (5,
infinity). The function is concave up on (3,4). Give a sketch of
the curve. (you do not need precise y-values. You need the correct
shape of the curve)

find the interval where the function is increasing and
decreasing f(x) =(x-8)^2/3
a) decreasing (8, infinity) increasing (-infinity, 8) local max
f(8)=0
a) decreasing (- infinity, 8) increasing (8, infinity) local max
f(8)=0
a) decreasing (-infinity, infinity) no extrema
a) increasing (-infinity, infinitely) no extrema

Let f be defined on the (0,infinity). Prove that the limit as x
approaches infinity of F(x) =L if and only if the limit as x
approaches 0 from the right of f(1/x) = L. Does this hold if we
replace L with either infinity or negative infinity?

1) Consider the function.
f(x) = x5 − 5
(a) Find the inverse function of f.
f −1(x) =
2)
Consider the function
f(x) = (1 + x)3/x.
Estimate the limit
lim x → 0 (1 + x)3/x
by evaluating f at x-values near 0. (Round
your answer to five significant figures.)
=

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 : 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.

Let f be a monotonic increasing function on a closed interval
[a, b]. Show that f is
Riemann integrable on [a, b].

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