Calculus 4.1 – Increasing/Decreasing Functions and Relative Extrema

Derivative & Tangent‐Line Slope

• The first derivative $f'(x)$ gives the slope of the tangent line to the curve at the point $(x,f(x))$.
• Sign of the derivative:
• f'(c)>0 \Rightarrow slope positive $\Rightarrow$ function rising (increasing) at $x=c$.
• f'(c)<0 \Rightarrow slope negative $\Rightarrow$ function falling (decreasing) at $x=c$.
• A change in sign of $f'(x)$ (from + to – or – to +) signals a switch between increasing and decreasing behaviour.

Increasing & Decreasing Intervals

• A function is
• increasing on any interval where f'(x)>0,
• decreasing on any interval where f'(x)<0.
• Graphical memory aid: wherever the derivative graph lies above the $x$-axis, the original function is climbing; below the axis it is falling.

Critical Numbers & Critical Points

• Critical number = an $x$–value satisfying either
$f'(x)=0 \quad \text{or} \quad f'(x)\;\text{does not exist (DNE)}$
provided $x$ is inside the domain of $f$.
• Critical point = the ordered pair $(c,f(c))$ where $c$ is a critical number.
• Relative extrema (local maxima or minima) can occur only at critical points, never in the interior of an interval where $f'(x)\neq0$.

6-Step Monotonicity Test (used repeatedly in examples)

1. Differentiate $f(x)$ to obtain $f'(x)$.

2. Solve $f'(x)=0$ and note any points where $f'(x)$ DNE $\Rightarrow$ list all critical numbers.

3. Order the critical numbers on a number line; pick convenient test $x$–values inside each sub-interval.

4. Evaluate the sign of $f'(x)$ at each test point.

5. Conclude:
   • f'(x)>0 \Rightarrow $f$ is increasing on that interval.
   • f'(x)<0 \Rightarrow $f$ is decreasing on that interval.

6. Optional: verify quickly with a graphing calculator.

Example 1

Function

$f(x)=x^3-9x^2-48x+52$

Derivative

$f'(x)=3x^2-18x-48=3(x^2-6x-16)=3(x-8)(x+2)$

Critical numbers

$x=-2,\;x=8$ (solve $f'(x)=0$; derivative exists everywhere).

Test‐point chart (sample points: $x=-5,0,10$)

• f'(-5)>0, f'(0)<0, f'(10)>0

Monotonicity

• Increasing on $(-\infty,-2)\cup(8,\infty)$
• Decreasing on $(-2,8)$

Relative extrema

• $x=-2$ changes + to – $\Rightarrow$ relative maximum at $(-2,f(-2))$.
• $x=8$ changes – to + $\Rightarrow$ relative minimum at $(8,f(8))$.

Example 2

Function

$f(x)=x^4-4x^3$

Derivative

$f'(x)=4x^3-12x^2=4x^2(x-3)$

Critical numbers

$x=0,\;x=3$ (from factorization; derivative exists everywhere).

Test points (e.g.

$x=-2,1,5$)
• x<0: f'(x)>0
• $03$: f'(x)>0

Conclusions

• Increasing on $(-\infty,0)\cup(3,\infty)$
• Decreasing on $(0,3)$
• Relative max at $x=0$; relative min at $x=3$.

Business Application: Cost, Revenue & Profit

Suppose production quantity $x$ (units).
• Cost: $C(x)=0.32x^2-0.00004x^3$
• Revenue: $R(x)=0.848x^2-0.00002x^3$
• Profit: $P(x)=R(x)-C(x)=0.528x^2+0.00002x^3$

Derivative of profit

$P'(x)=1.056x+0.00006x^2 = x(1.056+0.00006x)$

Critical numbers

$x=0$ and $x=-17{,}600$.
• Negative production has no business meaning, so only $0\le x\le3300$ appears relevant (interval supplied in transcript).

Profit behaviour

• Profit increasing on $00$ there).
• Past the practical domain end, further analysis required if x>3300.

Example 3 (Quadratic‐Rational Mix)

Function

$f(x)=0.002x^2+9+\frac{6912}{x}$
(Note: transcript shorthand “+6912x^{-1}”.)

Derivative

$f'(x)=0.004x-\frac{6912}{x^2}$
• Undefined at $x=0$ (division by 0) $\Rightarrow$ critical number.

Solve $f'(x)=0$

Multiply by $x^2$:
$0.004x^3-6912=0\;\Longrightarrow\;x^3=\frac{6912}{0.004}=1\,728\,000$
$x=120$ (real cube root).

Test sign of $f'(x)$

• For $0120$: pick $x=200$, f'(200)>0 $\Rightarrow$ increasing.

Relative extrema

• Derivative undefined at $x=0$ (boundary).
• $x=120$ switches – to + $\Rightarrow$ relative minimum at $(120,f(120))$.

Relative Maxima & Minima: Sign Chart Rule

• For a critical number $c$:
• If $f'(x)$ changes + → – across $c$ $\Rightarrow$ relative max at $c$.
• If $f'(x)$ changes – → + across $c$ $\Rightarrow$ relative min at $c$.
• Summary mnemonic: “positive to negative = peak, negative to positive = valley.”

Graphical Interpretation of f and f′

• When $f'(x)$ graph sits above the $x$-axis (f'(x)>0), the $f(x)$ graph is increasing; below the axis (f'(x)<0), $f(x)$ is decreasing.
• Critical numbers correspond to $x$-intercepts of $f'(x)$ or vertical asymptotes where $f'(x)$ is undefined.
• Relative extrema of $f$ appear at those $x$ values where the $f'(x)$ curve crosses the $x$-axis.

Example 4 (Polynomial Degree 3)

Function

$f(x)=x^3+6x^2+9x-8$

Derivative

$f'(x)=3x^2+12x+9=3\bigl(x^2+4x+3\bigr)=3(x+3)(x+1)$

Critical numbers

$x=-3,\;x=-1$.

Interval test (e.g. $x=-5, -2, 0$)

• x<-3: f'(x)>0
• $-3-1$: f'(x)>0

Results

• Increasing on $(-\infty,-3)\cup(-1,\infty)$.
• Decreasing on $(-3,-1)$.
• Relative max at $x=-3$; relative min at $x=-1$.

Additional Reminders & Exam Tips

• Quadratic‐factorization may fail; keep the quadratic formula handy:
$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
• Always verify that a derivative “DNE” point actually lies in the domain of $f$ before calling it critical.
• If the second derivative is available, $f''(c)\gtrless0$ can double-check whether $c$ is a max (concave down) or min (concave up), though the sign-change method remains conclusive.
• Combine analytic work with quick graph checks but never rely solely on calculator zoom; subtle extrema can hide.
• Label both the $x$-value (critical number) and the ordered pair (critical point) on homework or tests.

Quick Reference: Common Signs

• f(x)<0 $\Rightarrow$ graph below $x$-axis.
• f'(x)<0 $\Rightarrow$ slope negative $\Rightarrow$ decreasing interval.
• $f'(x)=0$ or undefined $\Rightarrow$ horizontal or vertical tangent/slope gap $\Rightarrow$ possible extremum.

Practice Checklist Before Exam

• Differentiate efficiently (power, product, quotient, chain rules).
• Build sign charts quickly and accurately.
• Translate between $f$ and $f'$ graphs.
• Apply concepts to business models (cost, revenue, profit).
• Recognize where real-world domains restrict $x$ (e.g.
negative production impossible).
• Show thorough work: derivative, critical numbers, sign test, final interval statement, extrema coordinates.