Let g:[0,1]->R be twice-differentiable (i.e both g and g' are differentiable functions) with g''(x)>0 for all x belong to [0,1].if g(0)>0 and g(1)=1 show that g(d)=d for some point d belong to (0,1) if and only if g'(1)>1.

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Proof:
"=>" Suppose g(d)=d for some d in (0,1). Since g"(x)>0 for all x in [0,1], then g'(x) is increasing. According to the Mean Value Theorem, we can ...

A-a function f:(a,b)->R is increasing on (a,b) if f(x)<=f(y) whenever x=0 for all x belong to (a,b).
b-show that the function g(x){x/(2+x^2 sin(1/x)) if x not=0 0 if x=0
is differentiable on R and satisfies g'(0)>0.Now

We have learned Rolle, Lagrange, Fermat, Taylor Theorems in our RealAnalysis class and we have finished differentiation. We just started integration. In this problem we are not supposed to use any material we haven't learned, ie integration. We are using the books by Rudin, Ross, Morrey/Protter.
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Let f(x) be integrable on [a,b], and let g(x) be nondecreasing and continuously differentiable on [a,b]. Let {p be element of P} be a partition of [a,b], and define
U(f,g,p) = SIGMA (Mi(g(the ith term of x) - g(the (i-1)th term of x))) as i=1 to n
L(f,g,p) = SIGMA (Ni(g(the ith term of x)-g(the (i-1)th term of x))) as i=1 t

These problems are realanalysis related problems. One of these is Liptuaz continuity/holder condition. It would be nice if you use mean value theorem for solving that problem.
1. Let f : [0, 2] ? R be continuous, assume that f is twicedifferentiable at all points of (a, b), and assume that f(0) = 0, f(1) = 1 and f(2) = 2. P

Suppose that f: [a,b] R is differentiable, that 0 < m f '(x) M for x є [a,b], and that f(a) < 0 < f(b). Show that the equation f(x) = 0 has a unique root in [a,b]. Show also that for any given x1 є [a,b], the sequence (xn), xn+1 = xn - for n = 1, 2,..., is well defined (i.e. for each n, xn є [a

1. Functions f, g, and h are continuous and differentiable for all real numbers, and
some of their values and values of their derivatives are given by the below table.
x f (x) g(x) h(x) f'(x) g'(x) h'(x)
0 1 -1 -1 4 1 -3
1 0 3 0 2 3 6
2 3 2

1.Consider the 2 functions f1(t) and f2(t);
1. f1(t) = { a1.e^ -2t for t>=0 and
= { 0 for t<0 }
f2(t) = { a2.e^ -t + a3.e^ -2t for t>=0 and
= { 0 for t<0 }
Find a1,a2 and a3 such that f1(t) and f2(t) are orthonormal on the interval 0 to infinity.
2. If Q is an n x n symmetric matrix and a1,a2 are such that 0 < a1I <