Application of Derivatives · Class 12 · CBSE · Science Mathematics
For f(x) = x³ − 6x² + 9x + 1, which statement about its local extrema is correct?
Practice question from Application of Derivatives (Class 12 · CBSE · Science Mathematics), with the correct answer and a step-by-step solution.
hard · MCQ · Q-E59343
For f(x) = x³ − 6x² + 9x + 1, which statement about its local extrema is correct?
- A.There is a local maximum at x = 3 and a local minimum at x = 1.
- B.There is a local maximum of 5 at x = 1 and a local minimum of 1 at x = 3.Correct answer
- C.There are no local extrema because f′(x) is never zero.
- D.There is a local maximum of 1 at x = 3 and a local minimum of 5 at x = 1.
Solution
f′(x) = 3(x − 1)(x − 3), so the critical points are x = 1 and x = 3. The derivative changes from positive to negative at x = 1, giving a local maximum f(1) = 5, and from negative to positive at x = 3, giving a local minimum f(3) = 1.
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