Functions of Several Variables — Question 7

Question 7 – Solution

Strategy Substitute the four samples one at a time.

Step 1: Coefficients From $q(0,0)=1$, $d=1$. Then $$a+d=6\Longrightarrow a=5,\qquad c+d=4\Longrightarrow c=3.$$ Finally, $$a+b+c+d=15\Longrightarrow 5+b+3+1=15\Longrightarrow b=6.$$ Thus $\boxed{q(x,y)=5x^2+6xy+3y^2+1}$.

Step 2: Evaluate $q(2,-1)=20-12+3+1=\boxed{12}$.

Step 3: Geometry Complete a square: $$5x^2+6xy+3y^2=5\left(x+\frac 35y\right)^2+\frac 65y^2.$$ Both coefficients are positive, so $q=k>1$ gives rotated ellipses, $q=1$ gives only the origin, and $q<1$ gives no real points.