Applications of the Derivative: Sign of the First Derivative, Increasing/Decreasing Intervals & Critical Numbers

Review – Increasing & Decreasing Functions in MATH 107 - Traditional (pre-calculus) criterion

  • Given two points (x1,f(x1)) and (x2,f(x2)) on the graph
  • If x1 < x2
    • If f(x1) < f(x2), then f is increasing on (x1,x2).
    • If f(x1) > f(x2), then f is decreasing on (x1,x2).
  • Purely relies on comparing $x$–values with their images; no derivatives involved.

Tangent Lines & Slope Intuition

  • Draw a tangent at a point (x0,f(x0)).
    • Slope of tangent = f'(x_0).
  • Geometric sign interpretation (recall from 107):
    • Rising line ( \ ): slope >0 → graph locally increasing.
    • Falling line ( // ): slope <0 → graph locally decreasing.
    • Horizontal line ( — ): slope =0 → graph locally constant.
    • Vertical line ( | ): slope undefined; indicates the derivative does not exist at that point.

Calculus Definitions (Derivative–Based)

Let f be differentiable on an open interval (a,b)\subset\text{Dom}(f).

  • Increasing on (a,b) ⟺ f'(x)>0 for every x\in(a,b).
  • Decreasing on (a,b) ⟺ f'(x)<0 for every x\in(a,b).
  • Constant on (a,b) ⟺ f'(x)=0 for every x\in(a,b).

Standard Procedure to Locate Increasing / Decreasing Intervals

  1. Compute f'(x).
  2. Solve f'(x)=0 and determine where f'(x) fails to exist
    → these $x$-values partition the real line ("test intervals").
  3. On each interval choose a test point and evaluate the sign of f'(x).
  4. Summarize:
    • f'(x)>0 → f is increasing there.
    • f'(x)<0 → f is decreasing there.

Example 1 — Polynomial

Given f(x)=x^{3}-6x+4\;(\text{domain }\Bbb R).

  1. f'(x)=3x^{2}-6.
  2. Solve 3x^{2}-6=0 \;\Rightarrow\; x=\pm\sqrt2.
  3. Sign chart for f'(x):
    • (-\infty,-\sqrt2) choose x=-2 → f'(x)>0.
    • (-\sqrt2,\sqrt2) choose x=0 → f'(x)<0.
    • (\sqrt2,\infty) choose x=3 → f'(x)>0.
  4. Conclusion
    • Increasing on (-\infty,-\sqrt2)\cup(\sqrt2,\infty).
    • Decreasing on (-\sqrt2,\sqrt2).

Example 2 — Quadratic (Classroom Exercise)

f(x)=x^{2}-3x+2 (domain \Bbb R)

  1. f'(x)=2x-3.
  2. 2x-3=0 \Rightarrow x=\tfrac32.
  3. Test intervals:
    • (-\infty,\tfrac32) → pick x=1 → f'(1)=-1<0.
    • (\tfrac32,\infty) → pick x=2 → f'(2)=1>0.
  4. Therefore
    • Decreasing on (-\infty,\tfrac32).
    • Increasing on (\tfrac32,\infty).

Critical Numbers – Formal Definition

Let f have domain D.
A number c\in D is a critical number of f if either
f'(c)=0 \quad\text{or}\quad f'(c)\text{ does not exist}. (Note: c must actually lie in the domain!)

Example 3 — Same Quadratic
  • f'(x)=2x-3; f'(\tfrac32)=0 and \tfrac32\in\Bbb R.
  • Critical number: c=\tfrac32.
Example 4 — Radical Function

f(x)=\sqrt{x-1}=(x-1)^{1/2}

  • Domain: x\ge1.
  • f'(x)=\dfrac{1}{2\sqrt{x-1}}.
  • No $x$ makes the derivative 0 (numerator is 1).
  • f'(1)=\dfrac{1}{0} (undefined) and 1 is in the domain.
  • Critical number: c=1.
Example 5 — Reciprocal Radical

f(x)=(x-1)^{-1/2}=\dfrac{1}{\sqrt{x-1}}

  • Domain: (1,\infty) (open at 1).
  • f'(x)=-\dfrac{1}{2}(x-1)^{-3/2}.
  • Derivative undefined at x=1, but 1\notin\text{Dom}(f).
  • Therefore no critical numbers for this function.

Additional Observations & Reminders

  • A vertical tangent (infinite slope) creates a point where f'(x) does not exist; it can supply a critical number only if the point lies inside the domain.
  • Constant functions have f'(x)=0 everywhere and are simultaneously increasing & decreasing by the derivative sign test.
  • When constructing sign charts, any convenient test value in an interval is valid ("pick –1 000 000 000 if you like"). Only the sign matters.
  • For radical expressions, secure the domain first; many pitfalls stem from forgetting to restrict to x\ge\text{(radicand \ge0)}.

Quick Checklist for Exams

  • [ ] State domains explicitly before hunting for critical numbers.
  • [ ] Isolate all f'(x)=0 solutions and all f'(x) DNE points.
  • [ ] Verify each