# Consider the function f : R2 -> R given by y f (I, y) = 1+ 12 + 92 restricted to the set S = {(x,y) ER? | y =x}, which we denote by f s. (a) [15 pts] Find all the critical points of f s. Warning:

Consider the function f : R2 -> R given by

y

f (I, y) = 1+ 12 + 92

restricted to the set S = {(x,y) ER? | y =x}, which we denote by f s.

(a) [15 pts] Find all the critical points of f s. Warning: Critical points of f s is not

the same as critical points of f. Hint: There are three critical points.

(b) [15 pts] Use the bordered Hessian to classify all the critical points in part (a).

(c) [5 pts] Compute the global maximum value of f g- Likewise, compute the global

minimum value of f s-

(d) [5 pts] In general, it is not true that a continuous function restricted to an un-

bounded set has either a global maximum value or a global minimum value, since

behavior of the function at infinity is never actually achieved. That being said,

explain why this is not a problem for us.

(e) [10 pts] Let D = {(x, y) ER? | y 2 12}, i.e. the region of the plane above and

including the parabola y = x2. Find both the global maximum value and the

global minimum value of f on D.

Remark: Be careful. It might help to use MATLAB to plot the graph of f.

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