# Extrema if you can

• Apr 14th 2008, 06:30 AM
dexza666
Extrema if you can
Hey i've found this one incredibly tough i don't know if it's just me(probably) but i think its hard and can't seem to get anywhere

Suppose f(x,y)= A(x2 + Bx + y2 +Cy). What values A, B, C give f(x,y) a local maximum value of 15 at the point (-2, 1)?

Also find all critical points of
f(x, y)=ex(1-cos y)

and classify these critical points.
• Apr 14th 2008, 08:15 AM
Peritus
$
\begin{gathered}
f(x,y) = A\left( {x^2 + Bx + y^2 + Cy} \right) \hfill \\
\hfill \\
\nabla f(x,y) = \left( {2Ax + AB} \right)\hat x + \left( {2By + AC} \right)\hat y = 0 \hfill \\
\end{gathered}
$

$
x = - \frac{B}
{2},y = - \frac{C}
{2}
$

we are told that the local maximum occurs at (-2,1) thus:

$
B = 4,C = - 2
$

next we are told that the value of the local maximum is 15 thus:

$
f(x,y) = \left. {A\left( {x^2 + Bx + y^2 + Cy} \right)} \right|_{(x,y) = ( - 2,1)} = 15
$

so A=-3.

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

I'm not sure what do you mean by: ex(1-cos y)
• Apr 14th 2008, 03:03 PM
dexza666
f(x,y) = e^x( 1 - cos(y) )
• Apr 16th 2008, 03:26 PM
mr fantastic
Quote:

Originally Posted by dexza666
[snip]
Also find all critical points of
f(x, y)=ex(1-cos y)

and classify these critical points.

Solve simultaneously:

$\frac{\partial f}{\partial x} = e^x (1 - \cos y) = 0$ .... (1)

$\frac{\partial f}{\partial y} = e^x \sin y = 0$ .... (2)

You get $(x, \, n \pi)$ where n is an integer. Now classify in the usual way.