Let be a continuous real valued function on satisfying the identity
Evaluate the integral
(D.M.Batinetu-Giurgiu , George Emil Palade)
Let be a twice differentiable function such that forall . If has a local extrema at and the value there is and it holds that
(i) Evaluate the area bounded by the graph of , the lines and the axis .
(ii) Let . If
holds then prove that
(iii) Prove that is invertible in . If you also know that the graph passes through the points and then evaluate the value of
(iv) Prove that forall it holds that
Since forall we conclude that is convex. We also conclude that is strictly increasing in . Since has a local extrema at this has to be a local minimum, because (it follows from Fermat’s theorem) and combining the monotony along with the root of the derivative we get that
(i) The area bounbed by the graph of , the lines and the axis is equal to
(ii) Successively we have
(iii) We begin by the classical change of variables thus:
(iv) We simply note that
and the result follows.
Let be positive numbers. Prove that
and we conclude the result.
Let denote the Euler’s Gamma function and denote the digamma function. Evaluate
The exercise can also be found at Aops.com .
Let denote the – th harmonic number and let . Prove that