Let f(x)= 2x +cos(2x)
(a) find the max value of f for 0 <(or equal to) x <(or equal to) pie.
(b) explain how the conditions of the mean value theorem are satisfied by f for 0 <(or equal to) x <(or equal to) pie. Find the value of x, whose existence is quaranteed by the mean value therom.
a. To find the max value between a closed interval, we have to test the critical values as well as the end points. To find the critical values, we see if the derivative crosses 0 at any point in the interval 0≤x≤π. Since there are no critical values in this interval, we only have to test the end points. So we plug in 0 and π in to the equation.
f(0) = 2(0) + cos(2(0))= 1
f(π)\textgreaterf(0) so the max value for f in the interval 0≤x≤π is at x= π which is f(π) = 1+2π
b. Mean Value theorem basically states that the slope of a line made between point a and point on the same function would appear as the derivate of at least one point between points a and b as long as the function is both continuous and differentiable.