Find the maximum profit that a company can make, if the profit function is given by :
`P(x)=41-24x-6x^(2).`

Answer

To find the maximum profit from the profit function \( P(x) = 41 - 24x - 6x^2 \), we will follow these steps:

### Step 1: Differentiate the Profit Function
We start by differentiating the profit function \( P(x) \) with respect to \( x \).

\[ P'(x) = \frac{d}{dx}(41 - 24x - 6x^2) \]

The derivative of a constant (41) is 0, the derivative of \(-24x\) is \(-24\), and the derivative of \(-6x^2\) is \(-12x\). Thus, we have:

\[ P'(x) = -24 - 12x \]

### Step 2: Set the Derivative Equal to Zero
To find the critical points, we set the derivative equal to zero:

\[ -24 - 12x = 0 \]

Solving for \( x \):

\[ -12x = 24 \\ x = -2 \]

### Step 3: Determine if it is a Maximum or Minimum
Next, we need to determine whether this critical point corresponds to a maximum or minimum. We do this by finding the second derivative \( P''(x) \).

\[ P''(x) = \frac{d}{dx}(-24 - 12x) = -12 \]

Since \( P''(x) = -12 \) is negative, it indicates that the function is concave down at \( x = -2 \), confirming that this point is a maximum.

### Step 4: Calculate the Maximum Profit
Now we substitute \( x = -2 \) back into the original profit function to find the maximum profit:

\[ P(-2) = 41 - 24(-2) - 6(-2)^2 \]

Calculating each term:

\[ = 41 + 48 - 6(4) \\ = 41 + 48 - 24 \\ = 65 \]

Thus, the maximum profit is:

\[ \text{Maximum Profit} = 65 \]

### Final Answer
The maximum profit that the company can make is \( \boxed{65} \).

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