Question

Let \( f(x) = 2x^3 - 9x^2 + 7. \) Which of the following is true?

• A. \( f \) is one-one in the interval \([-1, 1]\)
• B. \( f \) is one-one in the interval \([2, 4]\)
• C. \( f \) is NOT one-one in the interval \([-4, 0]\)
• D. \( f \) is NOT one-one in the interval \([0, 4]\)

Hint:

To check one-one nature, use the derivative test: if \( f'(x) \) doesn't change sign in an interval, \( f \) is one-one there.

Answer 1

The Correct Option is D

Step 1: Find the derivative.
\( f'(x) = 6x^2 - 18x = 6x(x - 3) \).

Step 2: Determine sign of \( f'(x) \).
\( f'(x) > 0 \) for \( x < 0 \) and \( x > 3 \); \( f'(x) < 0 \) for \( 0 < x < 3 \).

Step 3: Analyze intervals of monotonicity.
- \( f \) is increasing on \((-\infty, 0)\) and \((3, \infty)\).
- \( f \) is decreasing on \((0, 3)\).

Step 4: Determine where \( f \) is one-one.
In \([2, 4]\), \( f \) is strictly decreasing on \((0,3)\) and increasing on \((3,4]\), but since the turning point \( x=3 \) marks monotonic change, the function is one-one on \([2,4]\).

Final Answer: \( f(x) \) is one-one in \([2,4]\).