Problem text:

Turning points involving an exponential function and a constraint. I would like the solution please will explanation.
Thanks.
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Problem text:

Expressions involving datalists, together with a problem in logarithms. I would like to see the solutions please with some explanation. File is a jpeg. Please adjust to 25%. Thanks.

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Problem text:

Summation of a series, logarithms, drawing a graph.
I would like to see the solutions with explanation
please. File is a jpeg. Please adjust to 25% for best
view. Can wait 48 hours but 24 would be great.

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Problem text:

Need the formula for finding the arc length of the curves
generated by P as the cycloid moves right. P is a point
inside the circle but not all the way out to the
circumference.

Problem text:

Rewrite the following expression as a single logarithm with a coefficient of 1:


log(x^2-2x+1)-2log(x-1)

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Problem text:

Let x be an irrational number and y be a rational number. Show that these conditions are neither necessary nor sufficient for x to the power of y to be irrational.

Need a mathmatically aceptable proof using either direct/indirect/contradiction or existance proof.

Problem text:

given f(x)= sqrtx-2 and g(x)=1/x^2, find (g.f)(x) and its domain

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Problem text:

It can be proven that there are exactly three integer values of n for which 4^n + 4^2000 + 4^2003 is a perfect square (of an integer). Find these three values.

Problem text:

1.Find a monic polynomial p(x)=x^n+...of degree n with the least integral

¡Òp^2(x)dx on [-1;1]and find he minimal value of that integral

the same for ¡Òp^2(x)exp{-x^2}dx on (-¡Þ;¡Þ )
2.

let Kr:[-¦Ð:¦Ð]->R+
be non negative continuous function such that Kr(-¦Ð)=Kr(¦Ð)
1/(2¦Ð) ¡ÒKr(s)ds=1( integral on [-¦Ð:¦Ð])
and for any l>0
limKr(x)=0 when r->¡Þ
uniformly for |x|¡Ýl.let f be a 2 ¦Ð-periodic continuous function.Show that f*Kr->f for
r->¡Þ uniformaly with respect to x in [-¦Ð:¦Ð]

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Problem text:

The group of permutations on 4 symbols, S^, has 24 elements. For each order below,
describe at least one subgroup of S^ satisfying the given conditions or explain why such a subgroup does not exist.
a) order 3;
b) order 4, not cyclic
c) order 12;
d) order 9