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These are questions asked by Markiyan Hirnyk

Let us consider the maximum value of the polynomial

x^4+c*x^2+x^3+d*x-c-1

on the interval x=-1..1 as a function g of the parameters c and d. General considerations suggest its continuity. However, a plot3d of g does not confirm it. Also the question arises "Is the function g(c,d) bounded from below?". Here is my try with the DirectSearch and NLPSolve:

DirectSearch:-GlobalOptima( (a, b) -> g(a, b), variables = [a, b])

Error, (in Optimization:-NLPSolve) invalid input: PolynomialTools:-CoefficientVector expects its 1st argument, poly, to be of type polynom(anything, x), but received HFloat(HFloat(undefined))*x^4+HFloat(HFloat(undefined))*x^3+HFloat(HFloat(undefined))*x^2+HFloat(HFloat(undefined))*x+HFloat(HFloat(undefined))

Download bound.mw

I mean

J := int((x^2+2*x+1+(3*x+1)*sqrt(x+ln(x)))/(x*sqrt(x+ln(x))*(x+sqrt(x+ln(x)))), x);

Of course, with Maple.

How to solve the inequality

,

assuming a::real ?

Of course, with Maple. I'd like to demonstrate the difficulties, solving

>solve(log[2*abs(x-a)](abs(x+a)+abs(x-a)) < 1, x) assuming a > 0, a < 1/2

.

The correct answer under the above restrictions is

{x > 0, x < a} union {a < x, x < a + 1/2} union { -infinity < x, x < a - 1/2}.

This is a problem from Lviv math school olympiad '2016.

Here is my unsuccessful try

>restart; plots:-contourplot(exp(2*x/(x^2+y^2)), x = -2 .. 2, y = -2 .. 2,grid = [100, 100], coloring = [blue, red], contours = [.1, .3, .5, 1, 2]);

Of course, in Maple. Source http://www.mmf.lnu.edu.ua/

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