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VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 4 Theory

Applications of Derivatives: Optimization, Concavity & MVT

Critical points, inflection points, and Mean Value Theorem tangents

Move the secant interval [a, b] to observe the Mean Value Theorem guarantee: at least one point c exists where f'(c) perfectly parallels the secant slope.

1. Fermat's Theorem & The Mean Value Theorem

Local extrema can only occur at critical points where f'(x) = 0 or f' is undefined. The Mean Value Theorem states that if f is continuous on [a, b] and differentiable on (a, b), the curve's instantaneous velocity matches its average velocity at some intermediate time c.

f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b)-f(a)}{b-a}

Quick Reference & Cheat Sheet

Second Derivative Test
f′(x0)=0 and f′′(x0)>0  ⟹  Local Min (Concave Up)f'(x_0) = 0 \text{ and } f''(x_0) > 0 \implies \text{Local Min (Concave Up)}
Positive curvature creates an upward cup trapping a minimum.