Question

Let \( f(x) = a - (x-3)^{8/9} \), then the maxima of \( f(x) \) is:

Hint:

To find the maxima or minima of a function, check for critical points where the derivative is zero or undefined. In cases where the derivative never equals zero, analyze the behavior of the function to determine maxima or minima.

Answer 1

Step 1: Analyze the given function

The given function is: \[ f(x) = a - (x - 3)^{8/9} \] where \( a \) is a constant, and \( (x - 3)^{8/9} \) represents a fractional power of \( x - 3 \).

Step 2: Differentiate the function

To find the maxima or minima of \( f(x) \), we need to first find the derivative \( f'(x) \) and set it equal to zero. The derivative of \( f(x) = a - (x - 3)^{8/9} \) is: \[ f'(x) = - \frac{8}{9} (x - 3)^{-1/9} \]

Step 3: Find critical points

Set the derivative equal to zero to find critical points: \[ f'(x) = - \frac{8}{9} (x - 3)^{-1/9} = 0 \] This equation will never be equal to zero because \( (x - 3)^{-1/9} \) is never zero for any \( x \neq 3 \). Therefore, the function does not have any critical points where the derivative is zero.

Step 4: Check the behavior at \( x = 3 \)

The function involves a fractional power, so we need to check the behavior of \( f(x) \) near \( x = 3 \), where the term \( (x - 3)^{8/9} \) becomes zero. - For \( x = 3 \), we have: \[ f(3) = a - (3 - 3)^{8/9} = a - 0 = a \] - As \( x \to 3^+ \) (approaching 3 from the right), \( (x - 3)^{8/9} \) increases as \( x \) increases slightly from 3. - As \( x \to 3^- \) (approaching 3 from the left), \( (x - 3)^{8/9} \) also increases. Therefore, \( f(x) \) is decreasing for both \( x > 3 \) and \( x < 3 \), and the maximum occurs at \( x = 3 \).

The maximum value of \( f(x) \) occurs at \( x = 3 \), and the maximum value is \( f(3) = a \).