Class 12 Applications Of Derivatives · Long Answer3 Marksmonotonicity
Class 12 Maths · Applications Of Derivatives · Long Answer #55
Find the maximal intervals of strict increase and decrease of f(x)=3x⁴/2−4x³−45x²+51.
✅ Step-by-Step Solution
Final Answer: Increasing on [−3,0] and [5,∞); decreasing on (−∞,−3] and [0,5].
- Differentiate: f′=6x³−12x²−90x=6x(x+3)(x−5).
- The critical inputs are −3,0,5.
- The derivative signs on successive open intervals are −,+,−,+. Strict monotonicity extends to the finite critical endpoints.
- Increasing intervals are [−3,0] and [5,∞); decreasing intervals are (−∞,−3] and [0,5].
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Differentiate: f′=6x³−12x²−90x=6x(x+3)(x−5).
- [1 mark] The critical inputs are −3,0,5.
- [1 mark] The derivative signs on successive open intervals are −,+,−,+. Strict monotonicity extends to the finite critical endpoints.
- [0.5 marks] Increasing intervals are [−3,0] and [5,∞); decreasing intervals are (−∞,−3] and [0,5].
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