Let `f(x) = {(|x^(2)-2x| +a",",0 le x lt 5//2),(-2 x + 5",",x ge 5//2):}`. If f(x) has a maximum at x= 5/2, then the greatest value of |a| is

Answer

To solve the problem, we need to analyze the function \( f(x) \) defined as follows:

\[ f(x) = \begin{cases} |x^2 - 2x| + a & \text{for } 0 \leq x < \frac{5}{2} \\ -2x + 5 & \text{for } x \geq \frac{5}{2} \end{cases} \]

We need to find the greatest value of \( |a| \) such that \( f(x) \) has a maximum at \( x = \frac{5}{2} \).

### Step 1: Evaluate \( f\left(\frac{5}{2}\right) \)

First, we calculate \( f\left(\frac{5}{2}\right) \) using the second case of the function since \( x = \frac{5}{2} \) falls in the range \( x \geq \frac{5}{2} \):

\[ f\left(\frac{5}{2}\right) = -2\left(\frac{5}{2}\right) + 5 = -5 + 5 = 0 \]

### Step 2: Evaluate \( f\left(\frac{5}{2}^-\right) \)

Next, we need to evaluate \( f\left(\frac{5}{2}^-\right) \) using the first case of the function since \( \frac{5}{2}^- \) is slightly less than \( \frac{5}{2} \):

\[ f\left(\frac{5}{2}^-\right) = \left|\left(\frac{5}{2}\right)^2 - 2\left(\frac{5}{2}\right)\right| + a \]

Calculating \( \left(\frac{5}{2}\right)^2 - 2\left(\frac{5}{2}\right) \):

\[ \left(\frac{5}{2}\right)^2 = \frac{25}{4} \]
\[ 2\left(\frac{5}{2}\right) = 5 \]
\[ \left(\frac{5}{2}\right)^2 - 2\left(\frac{5}{2}\right) = \frac{25}{4} - 5 = \frac{25}{4} - \frac{20}{4} = \frac{5}{4} \]

Thus, we have:

\[ f\left(\frac{5}{2}^-\right) = \left|\frac{5}{4}\right| + a = \frac{5}{4} + a \]

### Step 3: Set up the condition for maximum

For \( f(x) \) to have a maximum at \( x = \frac{5}{2} \), we need:

\[ f\left(\frac{5}{2}^-\right) \leq f\left(\frac{5}{2}\right) \]

Substituting the values we calculated:

\[ \frac{5}{4} + a \leq 0 \]

### Step 4: Solve for \( a \)

Rearranging the inequality gives us:

\[ a \leq -\frac{5}{4} \]

### Step 5: Find the greatest value of \( |a| \)

The greatest value of \( |a| \) occurs when \( a = -\frac{5}{4} \). Therefore, we find:

\[ |a| = \frac{5}{4} \]

### Final Answer

The greatest value of \( |a| \) is:

\[ \boxed{\frac{5}{4}} \]