Class 12 Applications Of Derivatives · Very Long Answer5 Marksmonotonicity

Class 12 Maths · Applications Of Derivatives · Very Long Answer #105

Find the maximal strict monotonicity intervals of f(x)=x⁴−8x³+22x²−24x+21.

✅ Step-by-Step Solution

Final Answer: Increasing [1,2] and [3,∞); decreasing (−∞,1] and [2,3].

f′=4x³−24x²+44x−24.
Factor: f′=4(x−1)(x−2)(x−3); the critical inputs are 1,2,3.
The signs on (−∞,1),(1,2),(2,3),(3,∞) are −,+,−,+ respectively.
Strict monotonicity extends to each finite critical endpoint by continuity and the derivative signs in the interiors.
The maximal increasing intervals are [1,2] and [3,∞); the maximal decreasing intervals are (−∞,1] and [2,3].

[1 mark] f′=4x³−24x²+44x−24.
[1 mark] Factor: f′=4(x−1)(x−2)(x−3); the critical inputs are 1,2,3.
[1 mark] The signs on (−∞,1),(1,2),(2,3),(3,∞) are −,+,−,+ respectively.
[1 mark] Strict monotonicity extends to each finite critical endpoint by continuity and the derivative signs in the interiors.
[1 mark] The maximal increasing intervals are [1,2] and [3,∞); the maximal decreasing intervals are (−∞,1] and [2,3].

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