Differentiation Applications: Finding Extrema and Graphing

Recap: Derivative as Slope of a Tangent

  • Differentiation gives instantaneous rate of change or slope of the tangent line at any point on a curve.
  • Previously learned rules (power, product, quotient, chain) let us differentiate “a variety of functions.”
  • Key geometric fact: a tangent line is horizontal at the highest and lowest points of a smooth curve.
    • Horizontal tangent ⇒ slope =0=0 ⇒ derivative f′(x)=0f'(x)=0.
  • This single observation unlocks many applications, especially locating maxima/minima and solving optimization problems.

Extrema & Horizontal Tangents

  • Local (Relative) Maximum / Minimum:
    • Highest or lowest point in a neighborhood.
    • Function changes direction: increasing → decreasing (max) or decreasing → increasing (min).
  • Absolute (Global) Maximum / Minimum:
    • Highest or lowest value on the entire domain.
  • Test for smooth functions (no cusp or endpoint):
    • Step 1 – Compute derivative f′(x)f'(x).
    • Step 2 – Solve f′(x)=0f'(x)=0; solutions are called critical numbers.
    • Each critical number is a candidate for local/absolute extrema (additional tests needed to classify, e.g.
    • First‐Derivative Test, Second‐Derivative Test, or simply evaluating ff).

Types of Extrema in Representative Functions

  • Function with none: f(x)=x3f(x)=x^3 is strictly increasing ⇒ no local extrema.
  • Function with repeating absolute extrema: f(x)=sin⁡xf(x)=\sin x
    • Absolute/Local Max: f(x)=1f(x)=1 at x=π2+2πkx=\tfrac\pi2+2\pi k.
    • Absolute/Local Min: f(x)=−1f(x)=-1 at x=3π2+2πkx=\tfrac{3\pi}2+2\pi k.
  • Function with a finite set of local extrema: cubic f(x)=x3−3x2+1f(x)=x^3-3x^2+1.
    • Smooth, but changes direction only once up and once down.

Worked Example 1 – Polynomial f(x)=x3−3x2+1f(x)=x^3-3x^2+1

  1. Differentiate:
    f′(x)=3x2−6xf'(x)=3x^2-6x.
  2. Factor: f′(x)=3x (x−2)f'(x)=3x\,(x-2).
  3. Find critical numbers: solve 3x (x−2)=03x\,(x-2)=0.
    • x=0x=0 and x=2x=2.
  4. Classify (quick intuition—shape looks like an “S”):
    • x=0x=0 ⇒ local maximum.
    • x=2x=2 ⇒ local minimum.
  5. Exact extremal points:
    • f(0)=1f(0)=1 → point (0,1)(0,1).
    • f(2)=23−3⋅22+1=−3f(2)=2^3-3\cdot2^2+1=-3 → point (2,−3)(2,-3).

Worked Example 2 – Rational Function g(x)=xx2+1g(x)=\dfrac{x}{x^2+1}

  1. Use Quotient Rule:
    • Rule: ddx(uv)=v u′−u v′v2\dfrac{d}{dx}\left(\dfrac{u}{v}\right)=\dfrac{v\,u'-u\,v'}{v^2}.
    • Here u=xu=x, v=x2+1v=x^2+1, u′=1u'=1, v′=2xv'=2x.
    • g′(x)=(x2+1)(1)−x(2x)(x2+1)2g'(x)=\dfrac{(x^2+1)(1)-x(2x)}{(x^2+1)^2}
      =x2+1−2x2(x2+1)2=\dfrac{x^2+1-2x^2}{(x^2+1)^2}
      =−x2+1(x2+1)2=\dfrac{-x^2+1}{(x^2+1)^2}.
  2. Critical numbers from numerator only: solve −x2+1=0-x^2+1=0 ⇒ x2=1x^2=1 ⇒ x=±1x=\pm1.
  3. Function values:
    • g(1)=12g(1)=\tfrac{1}{2}.
    • g(−1)=−12g(-1)=\tfrac{-1}{2}.
    • Sketch shows a low valley at (−1,−0.5)(-1,-0.5) and a high peak at (1,0.5)(1,0.5) (mirrored across origin due to odd/even mix).
  4. Denominator never $=0$ ⇒ no vertical asymptotes; curve is smooth everywhere.

Implications for Graphing Higher‐Degree or Complicated Functions

  • Algebra alone (factoring, end behavior) provides x‐intercepts & limits but not exact turning‐point heights.
  • Calculus adds these intermediate “anchor points,” enabling near‐flawless sketches.
  • Recipe for a full graph:
    1. Determine domain, intercepts, asymptotes.
    2. Compute f′(x)f'(x), find critical numbers and classify.
    3. Evaluate f(x)f(x) at critical numbers → coordinates of extrema.
    4. Optionally, compute f′′(x)f''(x) to get concavity/inflection points.
  • Result: Accurate depiction of peaks, valleys, slopes—crucial for engineering design, economics, physics.

Connections to Future Topics – Optimization

  • Same procedure (critical numbers from f′=0f'=0) is the backbone of optimization problems: maximizing profit, minimizing cost, finding shortest path, etc.
  • Real‐world tie-in: Any time “best,” “least,” or “most” appears, expect to set derivative =0=0.

Practical & Philosophical Significance

  • Empowers us to predict and control systems—e.g. tuning a manufacturing process to minimize waste.
  • Shows how local information (slope) yields global conclusions (location of maximum).
  • Ethically, accurate modeling affects safety (bridges, medicine dosages). Misidentifying extrema could be catastrophic.

Study Tips & Next Steps

  • Practice differentiating varied functions (products, chains, quotients).
  • For each function you graph, consciously mark: intercepts, asymptotes, critical points, inflection points.
  • Compare first‐ and second‐derivative tests; understand when each is more efficient.
  • Preview optimization word problems to see the same math in context.
  • Review periodic functions to identify absolute vs. local extrema on restricted intervals.