1. **Problem statement:** A potter makes $x$ oval pots and $y$ bowl-shaped pots daily. 2. **Given:** - Cost per oval pot: $x + 20$ - Cost per bowl pot: $2y$ - Selling price per oval pot: 140 - Selling price per bowl pot: 160 3. **Goal:** Maximize profit by choosing $x$ and $y$. 4. **Profit function:** $$\text{Profit} = \text{Revenue} - \text{Cost} = (140x + 160y) - (x(x+20) + y(2y))$$ 5. **Simplify cost:** $$x(x+20) = x^2 + 20x$$ $$y(2y) = 2y^2$$ 6. **Profit function becomes:** $$P(x,y) = 140x + 160y - (x^2 + 20x + 2y^2) = -x^2 + 120x - 2y^2 + 160y$$ 7. **Find critical points by setting partial derivatives to zero:** $$\frac{\partial P}{\partial x} = -2x + 120 = 0 \implies 2x = 120 \implies x = 60$$ $$\frac{\partial P}{\partial y} = -4y + 160 = 0 \implies 4y = 160 \implies y = 40$$ 8. **Check second derivatives for maximum:** $$\frac{\partial^2 P}{\partial x^2} = -2 < 0$$ $$\frac{\partial^2 P}{\partial y^2} = -4 < 0$$ Both negative, so critical point is a maximum. 9. **Answer:** - Oval pots $x = 60$ - Bowl pots $y = 40$