How Many Solutions? Mastering f(x) = x for Convex Functions | JEE Advanced (Day 11/90)

Описание: Is f(x)=x solvable in 5 seconds? Day 11 explores how f''(x) greater than 0 changes everything. If you're struggling with the number of roots in Calculus, this geometry secret for JEE Advanced is a must-watch

Day 11 dives into the properties of Twice Differentiable Functions. This specific problem from the JEE Advanced (2023 Paper 2 Q7) tests your understanding of how the second derivative d^2f/dx^2 0 (concavity/convexity) limits the number of intersections a function can have with the line y = x.

Key Concepts Covered:
Convexity Analysis: Why d^2f/dx^2} 0 means the function is strictly "opening upwards."
Intersection Logic: Using Rolle’s Theorem to prove that a strictly convex function can intersect a line at most twice.
Existence vs. Universality: Breaking down the difference between "There exists" (Options A & C) and "For every" (Option B).
Graphing the Counter-examples: Visualizing functions where X_f = 0, 1, or 2 to eliminate incorrect statements.

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