Absolute Extrema on Finite Closed Intervals
In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
f(x) = (2/3)x − 5, −2 ≤ x ≤ 3
117. Suppose that the second derivative of the function y = f(x) isy" =(x+1)(x-2).
For what x-values does the graph of f have an inflection point?
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
dv/dt = (1/2)sec t tan t, v(0) = 1
54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x² − 32√x
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.
2. y=x^4/4-2x^2+4
