Differential Curve Sketching, Extreme Values & Optimization

1. Local Extrema & Critical Numbers

2. Fermat's Interior Extremum Theorem

Proof for Local Maximum: Suppose $f$ has a local maximum at $c$. Then there exists $\delta > 0$ such that $f(x) \le f(c)$ for all $x \in (c - \delta, c + \delta)$.

• For $h \in (0, \delta)$ (approaching from right): $f(c + h) - f(c) \le 0 \implies \frac{f(c + h) - f(c)}{h} \le 0 \implies f'_+(c) \le 0$.
• For $h \in (-\delta, 0)$ (approaching from left): $f(c + h) - f(c) \le 0 \implies \frac{f(c + h) - f(c)}{h} \ge 0 \implies f'_-(c) \ge 0$.

Since $f$ is differentiable at $c$, $f'(c) = f'_+(c) = f'_-(c)$. The only real number satisfying both $f'(c) \le 0$ and $f'(c) \ge 0$ is $f'(c) = 0 \quad \blacksquare$

3. The Closed Interval Method for Absolute Extrema

To find the absolute maximum and minimum of a continuous function $f$ on a closed bounded interval $[a, b]$:

1. Find all critical numbers $c_i \in (a, b)$.
2. Evaluate $f(c_i)$ at every critical number.
3. Evaluate $f(a)$ and $f(b)$ at the endpoints.
4. The greatest evaluated value is the absolute maximum; the least is the absolute minimum.

1. The First Derivative Test for Local Extrema

Let $c$ be a critical number of a continuous function $f$:

• If $f'(x)$ changes from positive to negative as $x$ increases through $c$, then $f(c)$ is a local maximum.
• If $f'(x)$ changes from negative to positive as $x$ increases through $c$, then $f(c)$ is a local minimum.
• If $f'(x)$ does not change sign across $c$ (e.g. $+ \to +$ or $- \to -$), then $f(c)$ is neither a local maximum nor a local minimum (e.g. $f(x) = x^3$ at $c = 0$).

2. Concavity & Inflection Points

Point of Inflection: A point $(c, f(c))$ on the curve where the function is continuous and the concavity changes (from upward to downward, or vice versa). A necessary condition for an inflection point is $f''(c) = 0$ or $f''(c)$ undefined.

3. The Second Derivative Test for Local Extrema

Suppose $f''(x)$ is continuous near a stationary point $c$ with $f'(c) = 0$:

• If $f''(c) > 0$, then the curve is concave upward at $c$, so $f(c)$ is a local minimum.
• If $f''(c) < 0$, then the curve is concave downward at $c$, so $f(c)$ is a local maximum.
• If $f''(c) = 0$, the test is inconclusive (e.g. $x^4$ has a minimum, $-x^4$ has a maximum, $x^3$ has an inflection point); one must revert to the First Derivative Test.

The Universal 7-Step Curve Sketching Protocol

To sketch the graph of an arbitrary function $y = f(x)$ with analytical precision without guesswork:

1. Step 1: Domain & Symmetries — Identify $\operatorname{Dom}(f)$. Test for even symmetry $f(-x) = f(x)$, odd symmetry $f(-x) = -f(x)$, or periodicity $f(x+T) = f(x)$.
2. Step 2: Intercepts — $y$-intercept at $(0, f(0))$. $x$-intercepts by solving $f(x) = 0$.
3. Step 3: Asymptotes — Vertical asymptotes where $Q(x) = 0$ with $\lim |f(x)| = \infty$. Horizontal asymptotes $\lim_{x \to \pm\infty} f(x) = L$. Slant asymptotes $y = mx + b$ if $\lim [f(x) - (mx+b)] = 0$.
4. Step 4: First Derivative $f'(x)$ — Critical numbers ($f'(x) = 0$ or undefined). Sign chart for $f'(x)$ to determine intervals of increase and decrease.
5. Step 5: Local Extrema — Apply First or Second Derivative Test to classify each critical point.
6. Step 6: Second Derivative $f''(x)$ — Determine $f''(x) = 0$ or undefined. Sign chart for $f''(x)$ to determine intervals of concavity upward/downward and exact inflection points.
7. Step 7: Global Assembly & Plot — Plot intercepts, asymptotes, extrema, and inflection points; sketch the smooth curve following the concavity and monotonicity signatures.

1. Applied Mathematical Optimization Protocol

1. Variable Identification & Sketch: Assign variables to all quantities. Draw a diagram.
2. Objective Function: Write an explicit formula for the quantity $Q$ to be maximized or minimized (e.g. volume, area, cost, material).
3. Constraint Equations: Relate secondary variables via geometric or physical constraints (e.g. surface area budget, perimeter, Pythagorean theorem) to express $Q = f(x)$ purely in terms of a single independent variable.
4. Domain Specification: Establish the feasible physical domain $x \in [a, b]$ or $(a, b)$.
5. Extremum Determination: Compute $f'(x) = 0$. Use the Closed Interval Method or Second Derivative Test to verify that the critical point is indeed the global optimum.

2. The Systematic Related Rates Protocol

When physical variables are related by an equation $F(x, y, z) = 0$ and change over time $t$:

1. Identify the given rates (e.g. $\frac{dx}{dt}$) and the target rate (e.g. $\frac{dy}{dt}$) at a specific instant $t_0$.
2. Formulate a geometric equation relating the static variables (never substitute numerical values that change with time before differentiating!).
3. Differentiate implicitly with respect to time $t$ using the Chain Rule: $\frac{d}{dt}[y^2] = 2y \frac{dy}{dt}$.
4. Substitute the instantaneous values and solve algebraically for the desired rate of change.