Increasing / Decreasing Functions, Critical Numbers & Profit Optimization

Increasing vs. Decreasing Functions

• Intuitive idea

  • Trace a curve left → right.

  • $y$-values go up ➔ function increases.

  • $y$-values go down ➔ function decreases.

  • Flat segment ➔ constant / neither.

• Graph-based check

  • Pick any horizontal slice: moving right, if heights rise → increasing; if fall → decreasing.

  • “Drive-a-car” metaphor:

  • Must push uphill → increasing (positive slope).

  • Car sits level → slope $=0$ (constant).

  • Rolls downhill → decreasing (negative slope).

• Slope / derivative viewpoint

  • For an open interval $I$ where $f$ is differentiable:

  • f'(x)>0\;\forall x\in I ⇒ $f$ is increasing on $I$.

  • f'(x)<0\;\forall x\in I ⇒ decreasing on $I$.

  • $f'(x)=0\;\forall x\in I$ ⇒ constant on $I$ (flat road).

Critical Numbers & Critical Points

• Critical number $c$ of $f$:

  1. $f'(c)=0$ (horizontal tangent) OR

  2. $f'(c)$ does not exist (sharp corner, cusp, vertical tangent)

• Critical point: ordered pair $(c,\,f(c))$.

• Because differentiability implies continuity, jumps cannot occur at interior critical numbers (they’d destroy the derivative).

• Every local max/min (extremum) occurs at

  • a critical number or

  • an endpoint of the domain.

  • ❗ Not every critical number is an extremum (e.g.
    $f(x)=x^{3}$ at $x=0$: flat but no max/min).

Procedure: Determine Increasing / Decreasing Intervals (First-Derivative Test)

1. Find $f'(x)$.

2. Solve $f'(x)=0$ and locate non-existent derivative points → list of critical numbers.

3. Add any domain breakpoints (where $f$ undefined) to number line.

4. Create open intervals between successive marks.

5. Choose a test $x$ inside each interval, plug into $f'(x)$ (sign only often suffices).

6. Record sign pattern:

   • $+$ = increasing, $-$ = decreasing.

7. Interpret

   • Sign changes $+\to-$ at $c$ ⇒ relative max.

   • Sign changes $-\to+$ at $c$ ⇒ relative min.

   • Same sign on both sides ⇒ no extremum at $c$ (maybe plateau or vertical tangent).

Why the test works (visual)

• $+$ then $-$: slope rises, flattens, then falls → hilltop.

• $-$ then $+$: slope falls, flattens, then rises → valley.

Sketching from Derivative Table

• Plot critical points on axes ($x$ coordinate from step 2, $y=f(x)$).

• Draw trend arrows between points using sign pattern.

• Connect smoothly:

  • Up-bowls connect decreasing→increasing (min).

  • Cap-shapes connect increasing→decreasing (max).

• Vertical tangents / corners: use sharp turn symbol.

Worked Symbolic Example 1

Function: $f(x)=-x^{3}-2x^{2}+15x+7$

1. $f'(x)=-3x^{2}-4x+15$.

2. Set $f'(x)=0$:
   $-3x^{2}-4x+15=0 \Longrightarrow 3x^{2}+4x-15=0$
   Factor $\to (3x-5)(x+3)=0$ ⇒ $x=-3,\;x=\tfrac{5}{3}$.

3. Intervals: $(-\infty,-3),\;(-3,\tfrac53),\;(\tfrac53,\infty)$.

4. Test points $x=-4,0,2$.

   • $x=-4$: product $(-)(-)(-)=-$ ⇒ decreasing.

   • $x=0$: $(-)(-)(+)=+$ ⇒ increasing.

   • $x=2$: $(-)(+)(+)= -$ ⇒ decreasing.

5. Pattern $-\; +\; -$:

   • $x=-3$: $-\to+$ ⇒ relative min.

   • $x=\tfrac53$: $+\to-$ ⇒ relative max.

6. Evaluate $f(-3)=-26$, $f(\tfrac53)=\tfrac{670}{27}$.

7. Increasing on $(-3,\tfrac53)$; decreasing on $(-\infty,-3)\cup(\tfrac53,\infty)$.

Worked Symbolic Example 2 (Only sign pattern shown)

• Critical numbers: $-3,1$.

• Sign table gives $+,-,+$.

  • Increasing on $(-\infty,-3)$ and $(1,\infty)$.

  • Decreasing on $(-3,1)$.

  • Max at $x=-3$, min at $x=1$.

Relating to Graph Pictures

• Mark turning points (red) where $f'(x)=0$ or DNE.

• Shade increasing arcs green, decreasing arcs orange.

• Plateaus or vertical tangents labelled “neither.”

• Corners still qualify as critical (derivative DNE).

Extrema Terminology

• Relative / local extrema: maxima or minima observed within a neighborhood.

• Absolute / global extrema (next lecture) require comparing entire domain.

• Endpoints may serve as local extrema if comparison set is one-sided.

No-Extremum Criticals

• Example $f(x)=x^{3}$: $f'(0)=0$ but $f$ rises on both sides (pattern $+,+$).

• Example $f(x)=-x^{3}$: pattern $-,-$.

• Conclusion: horizontal plateau, not extremum.

Business Application: Maximizing Profit

• Let

  • $C(q)$ = cost of producing $q$ units.

  • $p(q)$ = price (demand) per unit when $q$ units sold.

  • Revenue $R(q)=q\,p(q)$.

  • Profit $P(q)=R(q)-C(q)$.

• To maximize weekly profit:

  1. Find $P'(q)$.

  2. Solve $P'(q)=0$ for critical quantities.

  3. First-derivative test to decide max.

Bicycle Example

Given

• $C(q)=100+10q$

• $p(q)=50-2q$ (demand)

1. $R(q)=q(50-2q)=50q-2q^{2}$.

2. $P(q)=R(q)-C(q)=50q-2q^{2}-(100+10q) = -2q^{2}+40q-100$.

3. $P'(q) = -4q + 40$.

4. $P'(q)=0\Rightarrow -4q+40=0 \Rightarrow q=10$.

5. Test:

   • $q=0$ ⇒ P'(0)=40>0 (increasing).

   • $q=11$ ⇒ P'(11)=-4<0 (decreasing).

   • Pattern $+\to-$ ⇒ relative max at $q=10$.

6. Max profit $P(10)=-2(10)^{2}+40(10)-100 = 100$ dollars/week.

7. Optimal price: $p(10)=50-2(10)=30$ dollars/bike.

8. Recommendation: produce & sell 10 bikes/week at \$30 each → profit \$100/week.

• Graphically: widest gap between revenue curve (blue) & cost curve (red) at $q=10$.

Practical / Ethical Notes

• Economics: Knowing profit-max quantity prevents over-production (waste) or under-supply.

• Engineering: Identifying peak stress points requires local max analysis.

• Data science: Gradient sign changes pinpoint turning points in loss functions.

Common Pitfalls & Tips

• Forgetting to include points where $f$ is undefined in interval split.

• Mixing up $x$-intervals (answers are $x$-ranges, not $y$-ranges).

• Dropping the negative sign when factoring $f'(x)=0$.

• Calling every critical point an extremum – always run the sign test.

• Graph quickly by:

  1. Critical points.

  2. Sign arrows.

  3. Sketch arcs; no heavy algebra needed for rough shape.

Connections & Preview

• Second-derivative test: alternate (concavity-based) method for extrema.

• Implicit differentiation (next lecture, HW 6B) extends derivative tools to non-explicit $y$.

• Absolute extrema require evaluating endpoints plus all critical points ⇒ upcoming.