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Let n be a positive integer. Find the continuous functions f:ℝ->ℝ with the property that integral from 1 to x of f(ln(t)) dt=x^n ln(x) for all positive real numbers x.

Differentiating the integral equation with respect to x we obtain: f(ln(x))=nx n-1 ln(x)+x n-1 =x n-1 =x n-1 (nln(x)+1) , for any positive real number x.Let ln(x)=t , thus x=e t and f(t)=e t(n-1) (nt+1) , fo...
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What is the difference between a Supremum and a Maximum of a sequence?

A supremum is a number such that it is larger than all numbers in the given sequence. Although this is true for a maximum, the maximum has an additional property that it must be a member of the sequence so a...
BR
Answered by Ben R. • Maths tutor
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What does it mean for a matrix to be singular?

A singular matrix is one which has a determinant of zero. This has several important consequences depending on the context in which the matrix is being used: Firstly, it implies that the matrix is not invert...
JK
Answered by James K. • Maths tutor
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Solve the simultaneous equations y – 3x + 2 = 0, y^(2) – x – 6x^(2) = 0

Start on the easiest equation and solve to find yy-3x+2=0 <=> y=3x-2Then susbtitute in equation the expression of y in the second equation and simplify(3x-2) 2 -x-6x 2 =0 <=> 9x 2 -12x+4-x-6x 2 =...
MP
Answered by Maria P. • Maths tutor
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Solve the quadratic equation (x^2 + 6x = -2) by completing the square.

If we knew (x 2 + 6x) to be the first two terms of a perfect square expression, but not the constant term that would follow, we can work it out. We would use the fact that the expression can be factored as s...
LT
Answered by Lindie T. • Maths tutor
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