Analytical Applications of Differentiation
What you must understand
Use $f'$ to find intervals of increase/decrease and critical points (where $f'=0$ or undefined). Use the first derivative test or second derivative test to classify extrema. Use $f''$ for concavity and inflection points.
Apply the Mean Value Theorem: if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, some $c$ has $f'(c)=\frac{f(b)-f(a)}{b-a}$. Recognize Rolle's Theorem as a special case.
Solve optimization problems: define the objective, express it in one variable using constraints, differentiate, and verify the extremum. Sketch curves using sign charts for $f'$ and $f''$, and analyze motion (increasing/decreasing position, speeding up when velocity and acceleration share sign).
Traps
- Calling a critical point an extremum without sign-change verification.
- Ignoring endpoints for absolute extrema on closed intervals.
- Misidentifying inflection points without a concavity change.
- Applying MVT without confirming differentiability on the open interval.
- Forgetting domain constraints in optimization.
Worked example
$f(x)=x^3-3x$. $f'(x)=3x^2-3=0$ gives $x=\pm1$. $f''(x)=6x$. At $x=1$, $f''>0$ so local minimum; at $x=-1$, $f''<0$ so local maximum. Increasing on $(-\infty,-1)\cup(1,\infty)$.
Rule of thumb: build a sign chart for $f'$ and $f''$; extrema need sign changes, inflection points need concavity changes.