Analytical Applications of Differentiation
Core ideas
Use derivatives to describe function behavior. Critical points occur where $f'=0$ or $f'$ is undefined. The first derivative test classifies extrema by sign change; the second derivative test uses concavity. Increasing where $f'>0$, decreasing where $f'<0$, concave up where $f''>0$. Inflection points require a change in concavity.
The Mean Value Theorem guarantees a point where the instantaneous slope equals the average slope on a closed interval, given continuity on the closed interval and differentiability on the open one. Extreme Value Theorem guarantees absolute extrema on a closed interval. Optimization problems require identifying a constraint and an objective, then testing critical points and endpoints. Curve sketching combines all of this with asymptotes and intercepts.
Traps
- Reporting a critical point as an extremum without verifying a sign change or second-derivative sign.
- Forgetting to check endpoints when the interval is closed; absolute extrema can occur there.
- Confusing inflection points (concavity change) with critical points (slope behavior).
- MVT hypotheses matter: continuity on the closed interval and differentiability on the open interval.
Worked example
For $f(x)=x^3-3x$ on $[-2,2]$: $f'(x)=3x^2-3=0$ gives $x=\pm1$. Evaluate $f(-2)=-2$, $f(-1)=2$, $f(1)=-2$, $f(2)=2$. The absolute maximum is $2$ (at $x=-1$ and $x=2$) and the absolute minimum is $-2$ (at $x=-2$ and $x=1$).