Mastering Functions & Extrema
Identify local peaks, global extrema, and critical points purely using derivative tests and analytical calculus techniques.
Critical Points & Local Extrema
Find stationary points and classify local maximums or minimums using First or Second Derivative tests.
Absolute Extrema on Closed Intervals
Apply the Extreme Value Theorem by testing interior critical points against interval endpoints.
Concavity & 2nd Derivative Test
Classify critical values instantly using the sign of the second derivative $f''(c)$.
Mixed Extrema Challenge
Randomized mix of polynomials, rational, and trigonometric functions across all concept types.
1st Derivative Test
• If $f'(x)$ changes $+ \to -$, local Maximum.
• If $f'(x)$ changes $- \to +$, local Minimum.
• If sign does not change, Neither.
2nd Derivative Test
• $f'(c) = 0$ and $f''(c) < 0 \implies$ Local Max (concave down).
• $f'(c) = 0$ and $f''(c) > 0 \implies$ Local Min (concave up).
• $f''(c) = 0 \implies$ Inconclusive.
Extreme Value Theorem (Closed Interval $[a,b]$)
Calculate $f(x)$ at all critical points in $(a,b)$ and endpoints $a$ and $b$. The highest output is the Absolute Maximum; the lowest output is the Absolute Minimum.