43. any one know CALC?

Question

at how many points on the interval [0,pi] does f(x) = 2inx+sin4x satisfy the mean value theorem???

Answer 1

Mean value theorem implies that you can draw a straight line (the 'slope' = the average derivative of the interval) from f(0) to f(pi).

But f(0) contains lnx, which is not defined at 0 (call it minus infinity). This makes the 'average slope' plus infinity, which is only matched by the true slope at x=0.

Changing the interval to (0, pi] will not help.

It would be a nice challenge on the interval [1,pi] or if f(x) = 2*ln(x+1) + sin(4x)