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Differential equation
Let be a twice differentiable function such that
If ,
find an explicit formula of
.
Solution
We have successively
Convex function
Let be a convex function on a convex domain
and
a convex non-decreasing function on
. Prove that the composition of
is convex on
.
Solution
We want to prove that for it holds that
We have:
Polynomial equation
Let denote the golden ratio. Solve the equation
Solution
First of all we note that
We easily note that is one root of the equation, hence using Horner we get that
Hence is a double root and the other root is
.
Similarity implies equality?
Let be similar to
. Does
hold?
Solution
No! Take then
. The matrices are similar but not equal.