导数取点

Solve high-school or college-entrance-exam style derivative problems centered on root existence, root count, and parameter ranges. Use when a problem asks whether a function has roots/intersections, exactly how many roots it has, or which parameters make roots exist. The skill analyzes with derivatives, monotonic intervals, extrema, limits, and value ranges, then rewrites the final proof using concrete point selection, sign changes, continuity, and the intermediate value theorem. In teaching mode, explain where chosen points come from using tangent bounds, exponential/log inequalities, Taylor-style estimates, quadratic fitting, homomorphic transformations, or explicit threshold solving.

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