Question

Find two level curve of the function: f: R^2→R defined by : f(x,y)= 4x^2+ 16y^2+8

Draw their rough sketches

Answer 1

$$f(x,y)=4x^2+16y^2+8$$

$$z=4x^2+16y^2+8$$

$z=c$

$c=8:$

$$4x^2+16y^2+8=8$$

$$4x^2+16y^2=0$$

$$Point (0,0)$$

$c=16:$

$$4x^2+16y^2+8=16$$

$$4x^2+16y^2=8$$

$$\dfrac{x^2}{(\sqrt{2})^2}+\dfrac{y^2}{(\dfrac{\sqrt{2}}{2})^2}=1, ellipse$$

$c=24:$

$$4x^2+16y^2+8=24$$

$$4x^2+16y^2=16$$

$$\dfrac{x^2}{(2)^2}+\dfrac{y^2}{(1)^2}=1, ellipse$$