Extrema if you can

• Apr 14th 2008, 05:30 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, 07:15 AM
Peritus
$\displaystyle \begin{gathered} f(x,y) = A\left( {x^2 + Bx + y^2 + Cy} \right) \hfill \\ \hfill \\ \nabla f(x,y) = \left( {2Ax + AB} \right)\hat x + \left( {2By + AC} \right)\hat y = 0 \hfill \\ \end{gathered}$

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

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

$\displaystyle B = 4,C = - 2$

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

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

so A=-3.

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I'm not sure what do you mean by: ex(1-cos y)
• Apr 14th 2008, 02:03 PM
dexza666
f(x,y) = e^x( 1 - cos(y) )
• Apr 16th 2008, 02:26 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:

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

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

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