I'm practicing toward final exam and I'm having difficulty to solve particular question:
Given function f(x)=x^9 / sqrt(1+x) defined on closed interval [0,1]
I need to prove without calculating the integral that the definite integral between 0 and 1 with f(x) as integrand is at least 1/(10*sqrt(2)) and at most 1/10.
Our course material was only about Darboux sums (not Riemann)
My idea was to find a certain partition of [0,1] interval and show that for upper and lower integral Darboux sums hold 1/(10*sqrt(2)) <= s(P)<=S(P) <= 1/10
I also noticed that the desired estimate is quite good so I probably need a very fine partition to be able to prove.
Is it a correct way of thinking to solve the problem?
If yes, how can i think of a way of constructing the desired partition ?