If `f(x)=a-(x-3)^(8//9)`, then the maximum value of `f(x)` is

Options

A. 3
B. `a-3`
C. a
D. none of these

Answer

To find the maximum value of the function \( f(x) = a - (x - 3)^{\frac{8}{9}} \), we can follow these steps:

### Step 1: Analyze the function
The function is given as:
\[ f(x) = a - (x - 3)^{\frac{8}{9}} \]
Here, \( (x - 3)^{\frac{8}{9}} \) is a non-negative term because any real number raised to an even power (in this case, \( \frac{8}{9} \)) is non-negative.

### Step 2: Determine the minimum value of \( (x - 3)^{\frac{8}{9}} \)
The expression \( (x - 3)^{\frac{8}{9}} \) reaches its minimum value when \( x - 3 = 0 \), which occurs at \( x = 3 \). At this point:
\[ (x - 3)^{\frac{8}{9}} = 0 \]

### Step 3: Substitute \( x = 3 \) into the function
Substituting \( x = 3 \) into the function gives:
\[ f(3) = a - (3 - 3)^{\frac{8}{9}} = a - 0 = a \]

### Step 4: Determine the behavior of \( f(x) \)
As \( x \) moves away from 3 (either increasing or decreasing), \( (x - 3)^{\frac{8}{9}} \) becomes positive, which means \( f(x) \) will decrease from its value at \( x = 3 \). Therefore, \( f(x) \) will be less than \( a \) for any \( x \neq 3 \).

### Step 5: Conclusion
Thus, the maximum value of \( f(x) \) occurs at \( x = 3 \) and is equal to \( a \).

\[ \text{Maximum value of } f(x) = a \]