In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
39. y = 8 / (x² + 4) (Witch of Agnesi)
The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an intensity eight times that of the other, are 6 m apart. How far from the stronger light is the total illumination least?
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local extreme values, if any, saying where they occur.
f(x) = x¹ᐟ³(x + 8)
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.
80. y' = 1 - cot²θ, for 0 < θ < π
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.
77. y' = cot(θ/2), for 0 < θ < 2π
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
dy/dx = 1/x² + x, x > 0; y(2) = 1
