To find the maximum of the function $z = f(x, y) = 43x + 14y + 70$ subject to the constraints $y \geq 0$ and $y \leq 16 - x^2$, we will proceed by checking the critical points and the boundary of the region defined by these constraints.

Step 1: Analyze the constraint $y \leq 16 - x^2$

The constraint $y \leq 16 - x^2$ defines a region bounded by the parabola $y = 16 - x^2$, and since $y \geq 0$, we are only interested in the part of the parabola where $16 - x^2 \geq 0$, or $x^2 \leq 16$, meaning $-4 \leq x \leq 4$.

Thus, we are working within the following region:

$y = 0 \quad \text{(the lower boundary)}, \quad y = 16 - x^2 \quad \text{(the upper boundary)}, \quad -4 \leq x \leq 4$

Step 2: Evaluate the function along the boundaries

Along $y = 0$:

When $y = 0$, the function simplifies to:

$z = 43x + 14(0) + 70 = 43x + 70$

Now, evaluate this for $x = -4$ and $x = 4$:

• At $x = -4$: $z = 43(-4) + 70 = -172 + 70 = -102$

• At $x = 4$: $z = 43(4) + 70 = 172 + 70 = 242$

Thus, on the lower boundary $y = 0$, the function reaches a minimum value of $-102$ at $x = -4$ and a maximum value of $242$ at $x = 4$.

Along $y = 16 - x^2$:

Now, substitute $y = 16 - x^2$ into the function:

$z = 43x + 14(16 - x^2) + 70 = 43x + 224 - 14x^2 + 70 = 43x - 14x^2 + 294$

We need to find the critical points by taking the derivative of this expression with respect to $x$ and setting it equal to zero:

$\frac{d}{dx}(43x - 14x^2 + 294) = 43 - 28x$

Setting $43 - 28x = 0$, we get:

$x = \frac{43}{28} \approx 1.54$

Now, evaluate $z$ at $x = 1.54$:

$y = 16 - (1.54)^2 \approx 16 - 2.37 = 13.63$

$z = 43(1.54) - 14(1.54)^2 + 294 \approx 66.22 - 33.17 + 294 = 327.05$

Thus, $z \approx 327.05$ at $x \approx 1.54$.

Step 3: Conclusion

The maximum value occurs along the boundary $y = 16 - x^2$ when $x \approx 1.54$, and the approximate maximum value is:

$f_{\max} \approx 327.05$

Summary:

• $f_{\min} = -102$ at $(x, y) = (-4, 0)$
• $f_{\max} \approx 327.05$ at $(x, y) \approx (1.54, 13.63)$

Would you like more details or have any questions?

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Related Questions:

1. How do we verify whether a critical point is a maximum or a minimum in constrained optimization?
2. Can we use Lagrange multipliers for this problem? How would that work?
3. What would the behavior of the function be if we changed the constants in the function $z = 43x + 14y + 70$?
4. How do we interpret the constraint $y \leq 16 - x^2$ geometrically?
5. Could we solve this problem using a graphical method?

Tip:

When solving optimization problems with constraints, always check the boundary conditions as the extrema often occur on these boundaries.