Applications of Derivatives
Application of Derivatives
nta_pyq_2025_jan
Grade 12
Question:
Let (2, 3) be the largest open interval in which the function f (x) = 2 log (x - 2) - x + ax + 1 is strictly e 2 increasing and (b, c) be the largest open interval, in which the function g(x) = (x - 1) (x + 2 - a) is strictly 3 2 decreasing. Then 100(a + b - c) is equal to :
Step-by-Step Solution
Key Concept: Apply the core result for monotonicity, tangents and extrema and simplify using the given constraints.
f (x) = ′ 2 - 2x + a \ge 0 x - 2 (2) -2 ′′ f (x) = - 2 < 0 2 (x - 2) ′ f (x) ↓ ′ f (3) \ge 0 2 - 6 + a \ge 0 a \ge 4 amin = 4 3 2 g(x) = (x - 1) (x + 2 - a) 3 2 g(x) = (x - 1) (x - 2) ′ 3 2 2 g (x) = (x - 1) 2(x - 2) + (x - 2) 3(x - 1) 2 = (x - 1) (x - 2)(2x - 2 + 3x - 6) 2 = (x - 1) (x - 2)(5x - 8) < 0 8 x \in ( , 2) 5 8 100(a + b - c) = 100 (4 + - 2) = 360 5
Correct Answer: 2