A trader realises 10x - x$^2$ Naira profit from the sale of x bags of corn. How many bags will give him the maximum profit?

1. Analyze the question

The profit function is given by $P(x) = 10x - x^2$. We need to find the value of $x$ (the number of bags) that results in the maximum profit. This is a classic optimization problem involving quadratic functions.

2. Explain the concept

In calculus, to find the maximum or minimum value of a function, we use the first derivative.

• If $P(x)$ is the profit function, the rate of change of profit with respect to the number of bags is $P'(x)$.
• At the maximum point, the slope of the curve is zero. Therefore, we set the first derivative to zero ($P'(x) = 0$) and solve for $x$.
• Alternatively, for a quadratic equation in the form $ax^2 + bx + c$, the maximum or minimum occurs at $x = -b / 2a$.

3. Show the solution

Method 1: Using Calculus (Differentiation) Given: $P(x) = 10x - x^2$

1. Find the derivative of $P(x)$ with respect to $x$: $P'(x) = \frac{d}{dx}(10x - x^2)$ $P'(x) = 10 - 2x$

2. Set the derivative to zero to find the stationary point: $10 - 2x = 0$ $10 = 2x$ $x = \frac{10}{2}$ $x = 5$

Method 2: Using the Vertex Formula For a quadratic equation $ax^2 + bx + c$: Here, $a = -1$ and $b = 10$. The maximum occurs at $x = \frac{-b}{2a}$ $x = \frac{-10}{2(-1)}$ $x = \frac{-10}{-2}$ $x = 5$

4. Identify the answer

Both methods confirm that the trader will make the maximum profit when he sells 5 bags of corn. This corresponds to option C.

5. Tip

Whenever you see a quadratic expression (where the highest power is 2) and you are asked to find the "maximum" or "minimum" value, you can always use the formula $x = -b/2a$ to save time during your JAMB exam. It is much faster than performing full differentiation!