5.9. LOCAL EXTREME POINTS 133
Therefore, f ′ (x) = 0. The case where x is a local maximum is similar. You just turn aroundthe inequality signs in the above.
Points at which the derivative of a function equals 0 are sometimes called critical points.Included in the set of critical points are those points where f ′ fails to exist. You could endup with a local maximum or minimum at such a point. Think of y = |x|. When x = 0 noderivative exists and it is a local minimum.
The following is a typical minimization problem, this one heavily dependent on geom-etry.
Example 5.9.3 Find the volume of the smallest right circular cone which can be circum-scribed about a ball of radius 4 inches. Such a cone has volume equal to π
3 r2h where r isthe radius of the cone and h the height.
Consider the following picture of a cross section in which l is the length of the linefrom the center of the ball to the to vertex of the cone as shown.
θ
4
l
The angle between the indicated radius of length 4 and the side of the cone is π/2 fromgeometric considerations. Thus l sinθ = 4 and the volume of the cone is
π
3(l +4)((l +4) tan(θ))2 .
2θ is no more than π and so θ < π/2. Thus tanθ is positive and equals sinθ√1−sin2 θ
=
4/l√1−(4/l)2 = 4√
l2−16. Then the volume of the cone is
π
3(l +4)
((l +4)
4√l2 −16
)2
=163
π
l2 −16(l +4)3
It seems there should be a solution to this problem and so we only have to find it by takinga derivative and setting it equal to 0 because the solution will surely be a local minimum.To take the derivative, use the rules of differentiation developed above. The derivative is− 16
3π
(l−4)2
(−l2 +8l +48
). Obviously you cannot have l = 4. Such a situation would not
even give a triangle. Therefore, the solution to the problem involves l2 − 8l − 48 = 0.There are two solutions, l = 12 or l = −4, the latter making absolutely no sense at all.Hence l = 12 must be the answer and the height of the cone is 16. The minimum volumeis then 16
3π
122−16 (12+4)3 = 5123 π. I think you probably could not do this problem without
the methods of calculus.Now here is an example about minimizing cost. It is another example which you could
not work without the methods of calculus.