The profit function P(x) of a firm, selling x items per day is given by P(x)= `(150-x)x-1625`. Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit

Answer

To solve the problem, we need to find the number of items \( x \) that maximizes the profit function \( P(x) = (150 - x)x - 1625 \). We will also find the maximum profit.

### Step 1: Write down the profit function
The profit function is given by:
\[ P(x) = (150 - x)x - 1625 \]

### Step 2: Differentiate the profit function
To find the maximum profit, we need to differentiate \( P(x) \) with respect to \( x \):
\[ P'(x) = \frac{d}{dx}[(150 - x)x - 1625] \]
Using the product rule, where \( u = 150 - x \) and \( v = x \):
\[ P'(x) = u'v + uv' = (-1)x + (150 - x)(1) \]
This simplifies to:
\[ P'(x) = -x + 150 - x = 150 - 2x \]

### Step 3: Set the derivative equal to zero
To find the critical points, set \( P'(x) = 0 \):
\[ 150 - 2x = 0 \]
Solving for \( x \):
\[ 2x = 150 \implies x = 75 \]

### Step 4: Determine if this point is a maximum
To confirm that this is a maximum, we can find the second derivative \( P''(x) \):
\[ P''(x) = \frac{d}{dx}[-2] = -2 \]
Since \( P''(x) < 0 \), this indicates that \( P(x) \) has a maximum at \( x = 75 \).

### Step 5: Calculate the maximum profit
Now, substitute \( x = 75 \) back into the profit function to find the maximum profit:
\[ P(75) = (150 - 75) \cdot 75 - 1625 \]
Calculating this:
\[ P(75) = 75 \cdot 75 - 1625 = 5625 - 1625 = 4000 \]

### Conclusion
The firm should manufacture **75 items** to achieve the maximum profit, which is **4000**.

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