The figure below shows the level curves of a function f(x,y) around a maximum or

Question

The figure below shows the level curves of a function f(x,y) around a maximum or minimum M. One of the points P or Q has coordinates (x1, y1) and the other has coordinates (x2, y2). Suppose b > 0 and c > 0. Consider the two linear approximations to f given by

f(x,y) a + b(x x1) + c(y y1)

f(x,y) k + m(x x2) + n(y y2)

(a) What is the relationship between the values of a and k?

i) a = k

ii) a > k

iii) a < k

iv) a = k

(b) What are the coordinates of point P?

(x1, y1) =

(x2, y2) =

(c) Is M a maximum or minimum?

(d) Is the sign of m positive or negative?

(e) Is the sign of n positive or negative?

Explanation / Answer

a) a = k, because on the same level curve, the value of the function is same.

b) P has the coordinate (x2,y2) because fx should be negative there.

c) M is minima because fy is positive both at P and Q.

d) m is negative because fx switches sign when it goes from Q(x1,y1) to P(x2,y2).

e) n is positive because level curve is symmetric with respect to y axis.

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