Locate all relative maxima, relative minima, and saddle points, if any.

Question

f(x,y) = 5y^2+xy+y+8x+6

f has a = a.) relative maximum, b.) relative minimum, c.) saddle point, d.) is inconclusive

at the point (?,?)

Answer 1

f(x,y) = 5y^2 + xy + y + 8x +6

The gradient is

(y + 8 10y + x + 1)

The Hessian is

0 1

1 10

There is only one critical point

From the gradient, from the first term, we see that y = -8. From the second term, 10y + x + 1 = 0

10(-8) + x + 1 = 0

x = 79

(79, -8) is our critical point

Note that no x or y appears in the Hessian.

The eigenvalues are solved from λ² - 10λ - 1 = 0 At this point, we know that we have a positive and negative eigenvalue, so this is a saddle point. (We can solve the quadratic if you like; λ = 5 ± √26; the 0 as the first element of the Hessian makes it trivial to the see that the unnormalized eigenvectors are

(1 5 + √26) and (1 5 - √26)

If you move in the direction of the eigenvectors from (79, -8), you can verify that this is a saddlepoint. (moving in the direction of the first eigenvector, you will see an increase as one moves away from the critical point; moving in the direction of the second eigenvector, you will see a decrease).