Question

Determine whether​ rolle's theorem applies to the function shown below on the given interval. if​ so, find the​ point(s) that are guaranteed to exist by​ rolle's theorem. ​f(x)equals=​x(xminus−44​)squared2​; ​[0,44​]

Answer 1

Since the function is continuous between x = 0 and x = 44 then Rolle's theorem applies here.

Differentiating

y' = x * 2(x - 44) + (x - 44)^2

y' = 3x^2 - 176x + 1936 = 0 (at a turning point).

solving we get x = 44 , 14.67

y" = 6x - 176 which is negative for x = 14.67 so this gives a maximum value for f(x)

This maximum is at the point (14.67, 12,619.85)

There is a minimum at ( 44,0)

These are the required points