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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-1ln(x)+x^n-1=x

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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 s...

Answered by Ben R. • Maths tutor

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How do you diagonalise a matrix?

A matrix is a representation of a linear map with respect to any basis of vectors of our choice. We often deal with matrices with respect to the standard cartesian basis, (1,0,0) (0,1,0) (0,0,1), but we c...

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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 inv...

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-6x²=0 <=>...

Answered by Maria P. • Maths tutor

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