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7. [8 POINTS] Let f: R → R be a strictly increasing function. Prove by way...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
6. [8 POINTS) Letbe a nonzero real number. Prove by way of contrapositive that if x+ irrational, then is irrational. is 7. 18 POINTS Consider a collection of closed intervals ( hal. = 1.2.3.... such that lim(b,- ) = 0 Prove by way of contradiction that there cannot be more than one real number contained in each of these intervals.
PROVE:
4. If f : R → R is a strictly increasing function, f(0) = 0, a > 0 and b > 0, then
(4) Let f R -R be a strictly conve:r C2 function and let 0 a) Write the Euler-Lagrange equation for the minimizer u.(x) of the following problem: minimize u subject to: u E A, where A- 0,REC1[0 , 1and u (0 a u(1)b) b) Assuming the minimizer u(a) is a C2 function, prove t is strictly convex
(4) Let f R -R be a strictly conve:r C2 function and let 0
a) Write the Euler-Lagrange equation for the minimizer u.(x)...
3. Let the function f be a real valued bounded continuous function on R. Prove that there is a solution of the equation f(x) = x, xER. Now choose a number a with f(a) > a and define the sequence (an) recursively by defining al = a and a叶1 = f(an), where n E N. If f is strictly increasing on R, show that (an) converges to a solution of the equation (0.1). This method for approximating the solution is...
A function f:R HR is said to be strictly increasing if f(x1) < f(12) whenever I] < 12. Prove: If a differentiable function f is strictly increasing, then f'(x) > 0. Then give counterexamples to show that the following statements are false, in general. (i) If a differentiable function f is strictly increasing, then f'(2) >0 for all 1. (ii) If f'(x) > 0 for all x, then f is strictly increasing -
1. Let f: R\{-1} → R, f() = 2+1 (a) Prove that f is not an increasing function on its domain, but its restrictions to intervals fl-20.-1) and fl(-1,00) are strictly increasing. (b) Find a codomain for fl-1.00) that makes the function bijective. Find the composi- tional inverse of our function. Sketch both our function and its inverse on the same set of axes.
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Question 4 (20 points) Let F: R R be any homogeneous polynomial function (with degree no less than one) with at least one positive value. Prove that the function f:Rn R, f(x) F(x) 1, defines on f-1(0) a structure of smooth manifold.
Let f(x) = 2x + 8/x +1
(a) Find the interval(s) where the function is increasing and
the interval(s) where it is decreasing. If the answer cannot be
expressed as an interval, state DNE (short for does not exist).
(b) Find the relative maxima and relative minima, if any. If
none, state DNE.
(c) Determine where the graph of the function is concave upward
and where it is concave downward. If the answer cannot be expressed
as an interval, use...