1. Find and classify all critical points of the function f(x, y) = 5xye^(-y^2) 2

Question

1. Find and classify all critical points of the function f(x, y) = 5xye^(-y^2)

2. Show that the function f(x, y) = x^2 + 4y^2 -4xy + 2 has a finite number of critical points and that D = 0 at each one

3. For a function of one variable, it is impossible for a continuous function to have, for example, two local maxima without a local minimum (or vice versa). However, for functions of two or more variables, such fucnitons exist. Given the function:

f(x,y) = -(x^2-1)^2-(x^2-x-1)^2,

show that (-1, 0, 0) and (1, 2 0) are critical points and that both are local maxima. (In fact, these two points are the only critical points for this function)

Explanation / Answer

2)

f(x, y) = x^2 + 4y^2 -4xy + 2

fx = partial der of f woth x
fx = 2x - 4y = 0
x = 2y

fy = 8y - 4x = 0
2y - x = 0
x = 2y

fxx = 2
fyy = 8
fxy = -4

Now, we have
D = fxx*fyy - fxy^2

D = 2*8 - (-4)^2

D = 16 - 16

D = 0

And thus proved that D = 0!

--------------------------------------------------------------------

1)
f(x, y) = 5xye^(-y^2)

fx = partial der of f with x
fx = 5ye^(-y^2) = 0
y = 0 , e^(-y^2) = 0
y = 0 is the only one cuz e^(-y^2) is exponential and aint ever 0

fy = partial der of f with y
fy = 5xe^(-y^2) - 10xy^2e^(-y^2) = 0
5xe^(-y^2) * (1 - 2y^2) = 0
x = 0 , y = 1/sqrt2 and y = -1/sqrt2

So, we have only (0,0) as the critical...

And clearly this is a local minima

----------------------------------------------------------------------

3)
Clearly the function written here
f(x,y) = -(x^2-1)^2-(x^2-x-1)^2 seems incorrect

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.