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

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 \( a \).