Question

Consider the function \( f(x) = x^{2/3} \cdot (6 - x)^{1/3} \). Which of the following statements is false?

• A. \( f \) is increasing in the interval \( (0, 4) \)
• B. \( f \) is decreasing in the interval \( (6, \infty) \)
• C. \( f \) has a point of inflection at \( x = 0 \)
• D. \( f \) has a point of inflection at \( x = 6 \)

Hint:

For functions involving roots or powers, always check their differentiability and behavior near boundary points and intervals.

Answer 1

The Correct Option is C

We are given the function \( f(x) = x^{2/3} \cdot (6 - x)^{1/3} \). To check the statements, we analyze the behavior of the function and its derivatives.

Statement 1: \( f \) is increasing in the interval \( (0, 4) \)
By calculating the first derivative, \( f'(x) \), we observe that \( f'(x) > 0 \) for \( x \in (0, 4) \), confirming that \( f \) is increasing in this interval.

Statement 2: \( f \) is decreasing in the interval \( (6, \infty) \)
The function \( f(x) \) is not defined for \( x > 6 \) because the term \( (6 - x)^{1/3} \) becomes complex. So this statement is invalid (false).

Statement 3: \( f \) has a point of inflection at \( x = 0 \)
The second derivative test confirms that there is a point of inflection at \( x = 0 \), so this statement is true.

Statement 4: \( f \) has a point of inflection at \( x = 6 \)
At \( x = 6 \), the function is not differentiable, and thus, it does not have a point of inflection at this point. Therefore, this statement is also false.

Conclusion:
The function is not defined beyond \( x = 6 \), so Statement 2 is logically incorrect. Therefore, Statement 2 is the false one among the four.