Partial Derivatives — Question 8

Question 8 – Solution

Strategy. Compute the partial derivatives symbolically, evaluate at $(1,1)$, and solve the resulting three equations.

Step 1: Equations $$q_x=2ax+by,\qquad q_y=bx+2cy.$$ Thus $$2a+b=7,\qquad b+2c=8,\qquad a+b+c=6.$$

Step 2: Solve From the first two equations, $$a=\frac{7-b}{2},\qquad c=\frac{8-b}{2}.$$ Substitution into $a+b+c=6$ gives $$\frac{7-b}{2}+b+\frac{8-b}{2}=\frac{15}{2},$$ which cannot equal $6$. Therefore the supplied data are inconsistent.

Conclusion. $$\boxed{\text{No constants }a,b,c\text{ satisfy all three conditions.}}$$

Verification. In fact, adding the two derivative equations gives $2(a+b+c)=15$, forcing $q(1,1)=15/2$, not $6$.