Let `f(x)=a-(x-3)^(8//9)` then greatest value of f(x) is

Options

A. 3
B. a
C. no maximum vlaue
D. non of these

Explanation

To find the greatest value of the function \( f(x) = a - (x - 3)^{\frac{8}{9}} \), we will analyze the function step by step.

### Step 1: Understand the function
The function is defined as:
\[ f(x) = a - (x - 3)^{\frac{8}{9}} \]
Here, \( a \) is a constant, and \( (x - 3)^{\frac{8}{9}} \) is a term that we need to analyze.

### Step 2: Analyze the term \( (x - 3)^{\frac{8}{9}} \)
The term \( (x - 3)^{\frac{8}{9}} \) is always non-negative because raising any real number to an even power (in this case, \( \frac{8}{9} \) is positive) results in a non-negative value.

### Step 3: Determine the minimum value of \( (x - 3)^{\frac{8}{9}} \)
The minimum value of \( (x - 3)^{\frac{8}{9}} \) occurs when \( x - 3 = 0 \), which gives:
\[ (x - 3)^{\frac{8}{9}} = 0^{\frac{8}{9}} = 0 \]
This means the minimum value of \( (x - 3)^{\frac{8}{9}} \) is 0.

### Step 4: Substitute the minimum value into \( f(x) \)
Substituting the minimum value of \( (x - 3)^{\frac{8}{9}} \) into \( f(x) \):
\[ f(x) = a - 0 = a \]

### Step 5: Determine the greatest value of \( f(x) \)
Since \( (x - 3)^{\frac{8}{9}} \) can only increase from 0, the maximum value of \( f(x) \) occurs when \( (x - 3)^{\frac{8}{9}} = 0 \). Therefore, the greatest value of \( f(x) \) is:
\[ \text{Greatest value of } f(x) = a \]

### Conclusion
The greatest value of \( f(x) \) is \( a \).

---