if ` 2x - 7 - 5x^2` has maximum value at ` x=a ` then a=

Options

A. `-(1)/(5)`
B. `(1)/(5)`
C. `(34)/(5)`
D. `-(34)/(5)`

Answer

To solve the problem, we need to find the value of \( a \) where the quadratic function \( f(x) = 2x - 7 - 5x^2 \) has its maximum value.

### Step-by-Step Solution:

1. **Identify the quadratic function**:
The given function can be rewritten as:
\[ f(x) = -5x^2 + 2x - 7 \]
Here, we can identify the coefficients: \( a = -5 \), \( b = 2 \), and \( c = -7 \).

2. **Determine the vertex**:
The maximum or minimum value of a quadratic function \( ax^2 + bx + c \) occurs at the vertex, which can be found using the formula:
\[ x = -\frac{b}{2a} \]
Since we are looking for the maximum value, we will use this formula.

3. **Substitute the values of \( a \) and \( b \)**:
Plugging in the values of \( a \) and \( b \):
\[ x = -\frac{2}{2 \times -5} \]

4. **Calculate the value**:
Simplifying the expression:
\[ x = -\frac{2}{-10} = \frac{2}{10} = \frac{1}{5} \]

5. **Conclusion**:
Since we have determined that the maximum value occurs at \( x = a \), we find that:
\[ a = \frac{1}{5} \]

Thus, the value of \( a \) is \( \frac{1}{5} \).