1. A company finds that its profit $P(x)$ in thousands of dollars from producing $x$ units of a product is given by the quadratic function $$P(x) = -5x^2 + 300x - 2000.$$ Find the number of units $x$ that maximizes the profit and calculate the maximum profit. 2. The cost $C(x)$ in dollars to produce $x$ items is modeled by the quadratic function $$C(x) = 0.01x^2 - 4x + 5000.$$ Determine the production level $x$ that minimizes the cost and find the minimum cost. 3. A retailer notices that the demand $D(p)$ for a product depends on the price $p$ in dollars and is given by $$D(p) = -20p^2 + 800p - 6000.$$ Find the price $p$ that maximizes demand and the maximum demand. 4. The revenue $R(x)$ in dollars from selling $x$ units of a product is given by $$R(x) = -2x^2 + 150x.$$ Calculate the number of units $x$ that maximizes revenue and the maximum revenue. 5. An investment's value $V(t)$ in thousands of dollars after $t$ years is modeled by $$V(t) = -3t^2 + 36t + 100.$$ Find the time $t$ when the investment reaches its maximum value and determine that value. Each problem uses the quadratic function formula $$f(x) = ax^2 + bx + c$$ where $a$, $b$, and $c$ are constants. Key rules: - If $a < 0$, the parabola opens downward and the vertex represents a maximum. - If $a > 0$, the parabola opens upward and the vertex represents a minimum. - The vertex $x$-coordinate is found by $$x = -\frac{b}{2a}.$$