5.2 Exercises on local optima

  1. Is x = 1 a local maximum or a local minimum of the function  f (x) = −x3 + 3x − 2, or neither?

  2. Find all the local maxima and minima (if any) of the following functions.
    1.  f (x, y) = −x2 + xy − y2 + 2x + y
    2.  f (x, y) = e2x − 2x + 2y2 + 3

  3. Find all the local maxima and minima and all the global maxima and minima, if any, of the following functions. For any extreme point that you find, give both the maximizer or minimizer (value of x) and the maximum or minimum (value of  f (x)).
    1.  f (x) = x2 + 3 on [−1, 1]
    2.  f (x) = x3 − 3x + 5 on [−3, 3]
    3.  f (x) = x + 1/x on [1/2, 2]
    4.  f (x) = (x − 2)6 on [0, 4].

  4. Find all the local maxima (if any) of the following functions. For each local maximum that you find, determine, if possible, whether it is a global maximum.
    1.  f (x, y) = (1/3)x3 + 2xy − 2y2 − 6x.
    2.  f (x, y) = 3xy − x3 − y3.

  5. Let  f (x, y) = (y − x2)(y − 2x2).
    1. Draw a figure showing the regions of the (x, y) plane at which this function has positive values and the regions at which it has negative values.
    2. Fix a number a and restrict attention to values of x and y for which x = ay. That is, for each number a, consider the function ga(y) =  f (ay, y). Show that for every value of a the point y = 0 is a local minimum of ga(y).
    3. Is the point (x, y) = (0,0) a local minimum of  f ?

  6. Find all the local maxima and minima and all the global maxima and minima, if any, of the following functions. For any extreme point that you find, give both the maximizer or minimizer (value of x) and the maximum or minimum (value of  f (x)).
    1.  f (x) = 3x3 − 5x2 + x on the interval [0, 1]
    2.  f (x) = 3x3 − 5x2 + x on the interval [0, 2]
    3.  f (x) = (x − 4)2 + 5 on the interval [−5, 5]

  7. Consider the function  f (x, y) = (x − 2)4 + (y − 3)4.
    1. Show that this function has a minimum at (x, y) = (2, 3) (without using any calculus).
    2. Find all the solutions of the first-order conditions.
    3. Is the Hessian of  f  positive definite at any solution of the first-order conditions?

  8. Find all the local maximizers and minimizers of the following functions.
    1.  f (x1, x2, x3) = x12 + 3x22 − 3x1x2 + 4x2x3 + 6x32.
    2.  f (x1, x2, x3) = 29−(x12 + x22 + x32).
    3.  f (x1, x2, x3) = x1x3 + x12 − x2 + x2x3 + x22 + 3x32.

[Solutions]