Question

the height of a certain hill (in feet) is given h(x,y)=10(2xy-3x^2-4y^2-18x+28y+12)
where,
y is the distance (in miles) north;
x is the distance east of a city;
a)where the top of the hill is located?
b)how high is the hill?


i do not know how to solve this thing.

Answer 1

this may require some muti-var calculus
let the top be \[P_\max(x_\max,y_\max)\]
\[\frac{ \partial h(x,y) }{ \partial x } = 0, \frac{ \partial h(x,y) }{ \partial y } = 0\]
if there are two or more solution, use second derivative to check.
\[ \frac{ \partial^2 h(x_\max,y_\max) }{ \partial x^2 } <0, \frac{ \partial ^2h(x_\max,y_\max) }{ \partial y^2 } < 0\]
if there are still two or more solution, choose the one that make h largest