Expression `15- 7x- 2x^2` will be maximum when x= ?

Answer

To find the value of \( x \) that maximizes the expression \( 15 - 7x - 2x^2 \), we can follow these steps:

### Step 1: Write down the expression
The given expression is:
\[ f(x) = 15 - 7x - 2x^2 \]

### Step 2: Differentiate the expression
To find the maximum value, we need to differentiate the expression with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(15) - \frac{d}{dx}(7x) - \frac{d}{dx}(2x^2) \]
Using the rules of differentiation:
- The derivative of a constant (15) is 0.
- The derivative of \( -7x \) is \( -7 \).
- The derivative of \( -2x^2 \) is \( -4x \).

So, we have:
\[ f'(x) = 0 - 7 - 4x = -7 - 4x \]

### Step 3: Set the derivative equal to zero
To find the critical points, we set the derivative equal to zero:
\[ -7 - 4x = 0 \]

### Step 4: Solve for \( x \)
Now, we solve for \( x \):
\[ -4x = 7 \\ x = -\frac{7}{4} \]

### Step 5: Conclusion
The value of \( x \) that maximizes the expression \( 15 - 7x - 2x^2 \) is:
\[ x = -\frac{7}{4} \]