Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.
8. y = 2cosx - √2x, -π≤x≤3π/2

10. Catching rainwater A 1125 ft^3 open-top rectangular tank with a square base x ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy.

a. If the total cost is c=5(x^2+4xy) + 10xy, what values of x and y will minimize it?

b. Give a possible scenario for the cost function in part (a).

Finding Extrema from Graphs

In Exercises 7–10, find the absolute extreme values and where they occur.

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

82. y' = sin t, for 0 ≤ t ≤ 2π

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dr/dθ = −π sin (πθ), r(0) = 0

Absolute Extrema on Finite Closed Intervals

In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.

g(θ) = θ³ᐟ⁵, −32 ≤ θ ≤ 1

Motion with constant acceleration The standard equation for the position s of a body moving with a constant acceleration a along a coordinate line is s = (a/2)t² + v₀t + s₀, where v₀ and s₀ are the body’s velocity and position at time t = 0. Derive this equation by solving the initial value problem

Differential equation: d²s/dt² = a

Initial conditions: ds/dt = v₀ and s = s₀ when t=0.