Yahoo Answers 4-27-14
- Michael Brown
Find maxima, minima, and analyze the concavity of the function: f(Ө) = (a/2) cos(Ө) - d cos(Ө), a>0, d>0 Explain. Interesting problem. In one case, you will get a flat line.. if a/2=d, no max, min, or concavity Otherwise... Critical points are where the derivative=0 f '(Ө)=(d-a/2)sin(Ө). That equals zero when Ө=z*pi where z is an integer. Concavity depends on the sign of the 2nd derivative and the inflection points are when the 2nd derivative=0 f "(Ө)=(d-a/2)cos(Ө). That equals zero when Ө=odd multiples of pi/2 (pi/2, 3pi/2, 5pi/2) But this is going to depend on the sign of (d-a/2)... If d>a/2 (or the coefficient is positive), the cos is positive from 0 to pi/2 (concave up), negative from pi/2 to 3pi/2 (concave down) and positive again from 3pi/2 to 2pi (concave up) The extremes at 0 and 2pi (and all even integers) will be minima The extremes at pi/2 and 3pi/2 (and all odd integers) will be maxima If d<a/2 (or the coefficient is negative) all this switches Adjust the sliders to see what this looks like...