Question

Let $ f(x) = a + \left( (x - 4) \right)^4 / 9 $, $\text{ then minima of } $ f(x) $\text{ is} $

• A. 4
• B. a
• C. a - 4
• D. None of these

Hint:

For minima problems involving power functions like \( (x - 4)^4 \), differentiate and solve for critical points to find the value at which the function is minimized.

Answer 1

The Correct Option is B

To analyze the function \( f(x) = a + (x - 4)^{4/9} \) and determine its extremum:

1. Compute the Derivative:
We first find the derivative of \( f(x) \):

\[ f'(x) = \frac{d}{dx}\left[a + (x - 4)^{4/9}\right] = \frac{4}{9}(x - 4)^{-5/9} \]

2. Identify Critical Points:
The derivative is undefined when the denominator becomes zero:

\[ (x - 4)^{-5/9} = \frac{1}{(x - 4)^{5/9}} \Rightarrow \text{undefined at } x = 4 \]
Thus, \( x = 4 \) is a critical point where the derivative does not exist.

3. Determine Extremum at Critical Point:
Evaluating the function at \( x = 4 \):

\[ f(4) = a + (4 - 4)^{4/9} = a + 0 = a \]
Since the function has a well-defined value here while its derivative is undefined, this represents a point of extremum.

4. Verify Minimum Value:
For all \( x \neq 4 \):

\[ (x - 4)^{4/9} > 0 \Rightarrow f(x) = a + \text{positive term} > a \]
Therefore, \( f(4) = a \) is indeed the minimum value of the function.

Final Answer:
The function \( f(x) \) has its minimum value at \( x = 4 \), which is \( a \).

Answer 2

The function given is: \[ f(x) = a + \left( \frac{(x - 4)^4}{9} \right). \] To find the minima, we need to first differentiate the function with respect to \( x \) and set the derivative equal to zero.
Step 1: Differentiate the function
\[ f'(x) = \frac{4}{9} \cdot (x - 4)^3. \] To find the critical points, set the derivative equal to zero: \[ \frac{4}{9} \cdot (x - 4)^3 = 0. \] This gives: \[ (x - 4)^3 = 0 \quad \Rightarrow \quad x = 4. \]
Step 2: Check if it's a minimum
Since the function \( f(x) = a + \frac{(x - 4)^4}{9} \) is always non-negative, it reaches its minimum when \( x = 4 \), as \( (x - 4)^4 = 0 \) at this point. Thus, the value of the function at \( x = 4 \) is: \[ f(4) = a. \] Therefore, the minima of the function is \( a \).