2

once again.

I wish to prove that $$\frac{2}{\pi} = \cos\left( \frac{\pi t}{2}\right) + \sin\left( \frac{\pi}{2} ( 1-t ) \right)$$

for some t in the interval $( 0, 1 )$ given the function

$$f( x, y ) = \sin(\pi x ) + \cos(\pi y ).$$

I was told that I could prove it using the mean value theorem, but I am not exactly sure how to use it in this case.

Thank you for your help ahead of time.

lalaman
  • 297
  • It should be $\sin$ instead of $\cos$. I have edited it. Thank you for mentioning it. – lalaman Feb 14 '14 at 05:16
  • That changes the context of my original comment, so I deleted that. As now written, this indicates that $ \ x \ = \frac{1-t}{2} \ , \ y \ = \ \frac{t}{2} \ . $ – colormegone Feb 14 '14 at 05:24
  • I'm not sure about using the Mean Value Theorem. The Intermediate Value Theorem can help: the function is continuous and we are considering a parameterized path from $ \ (\frac{1}{2},0) \ $ to $ \ (0, \frac{1}{2}) \ . $ What are the values of the function at the endpoints? Where is $ \ \frac{2}{\pi} \ $ in relation to those function values? – colormegone Feb 14 '14 at 05:43

0 Answers0