5. Graphical Applications of Derivatives

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = 3x⁵ - 25x³ + 60x on [-2,3]

The figure shows six containers, each of which is filled from the top. Assume water is poured into the containers at a constant rate and each container is filled in 10 s. Assume also that the horizontal cross sections of the containers are always circles. Let h (t) be the depth of water in the container at time t, for 0 ≤ t ≤ 10 . <IMAGE>

d. For each container, where does h' (the derivative of h ) have an absolute maximum on [0 , 10]?

71. Locate and identify the absolute extreme values of cos(ln x) on [1/2, 2]

133. Find the absolute maximum value of

f(x) = x^2 * ln(1/x)

and say where it is assumed.