6.4 Tangents, Normals, Increasing/Decreasing Functions & Extrema

Tangents, Normals, Increasing/Decreasing Functions & Extrema

The practical power of differential calculus lies in its geometric and optimization applications (FSc Part II). In the Pakistan ECAT, questions on tangents, normals, angles of intersection between curves, intervals of increase and decrease, and local/global extrema appear every year. Mastering the analytical criteria for these geometric features—and pairing them with second-derivative tests and optimization strategies—ensures rapid and reliable problem resolution.

---

1. Slopes and Equations of Tangent and Normal Lines

The Tangent Line

Let $y = f(x)$ be a differentiable curve. The slope $m$ of the tangent line at a point $P(x_0, y_0)$ on the curve is given by the derivative evaluated at that point:

$m = \left.\frac{dy}{dx}\right|_{(x_0, y_0)} = f'(x_0)$

Using the point-slope form, the equation of the tangent line is:

$y - y_0 = m(x - x_0) \implies y - y_0 = f'(x_0)(x - x_0)$

Special Tangent Cases

• Horizontal Tangent: If $f'(x_0) = 0$, the tangent line is horizontal with equation $y = y_0$.
• Vertical Tangent: If $f'(x_0) \to \pm\infty$ (or $\left.\frac{dx}{dy}\right|_{(x_0, y_0)} = 0$), the tangent line is vertical with equation $x = x_0$.

The Normal Line

The normal line to a curve at $P(x_0, y_0)$ is the straight line perpendicular to the tangent line at that point. If the tangent slope $m \neq 0$, the slope of the normal line $m_N$ satisfies the orthogonality condition $m \cdot m_N = -1$:

$m_N = -\frac{1}{m} = -\frac{1}{f'(x_0)}$

The equation of the normal line is:

$y - y_0 = -\frac{1}{f'(x_0)}(x - x_0)$

• If the tangent is horizontal ($m = 0$), the normal is vertical: $x = x_0$.
• If the tangent is vertical ($m \to \infty$), the normal is horizontal: $y = y_0$.

Lengths of Tangent, Normal, Subtangent, and Subnormal

The classical geometric lengths associated with a point $P(x_0, y_0)$ with slope $m = f'(x_0)$ are defined as follows:

• Subtangent Length: $ST = \left|\frac{y_0}{m}\right|$
• Subnormal Length: $SN = |y_0 \cdot m|$
• Tangent Segment Length: $PT = \left|\frac{y_0}{m}\right| \sqrt{1 + m^2}$
• Normal Segment Length: $PN = |y_0| \sqrt{1 + m^2}$

---

2. Angle of Intersection Between Two Curves

The angle of intersection $\theta$ between two intersecting curves $y = f(x)$ and $y = g(x)$ at their common point of intersection $P(x_0, y_0)$ is defined as the acute angle between their respective tangent lines at $P$.

Let $m_1 = f'(x_0)$ and $m_2 = g'(x_0)$ be the slopes of the tangent lines. Using the angle formula between two lines:

$\tan\theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$

Critical Geometric Conditions

1. Orthogonal Intersection (Perpendicular Curves): The curves intersect at right angles ($\theta = 90^\circ$) if and only if: $1 + m_1 m_2 = 0 \implies m_1 m_2 = -1$
2. Tangential Intersection (Curves Touch Each Other): The curves touch each other ($\theta = 0^\circ$) if and only if they share a common tangent line: $m_1 = m_2$

---

3. Increasing and Decreasing Functions (Monotonicity)

Formal Definitions

A function $f(x)$ defined on an interval $I$ is:

• Strictly Increasing on $I$ if for all $x_1, x_2 \in I$ with $x_1 < x_2$, $f(x_1) < f(x_2)$.
• Strictly Decreasing on $I$ if for all $x_1, x_2 \in I$ with $x_1 < x_2$, $f(x_1) > f(x_2)$.

Derivative Tests for Monotonicity

By the Mean Value Theorem, for a differentiable function on $(a, b)$:

1. If $f'(x) > 0$ for all $x \in (a, b)$, then $f(x)$ is strictly increasing on $[a, b]$.
2. If $f'(x) < 0$ for all $x \in (a, b)$, then $f(x)$ is strictly decreasing on $[a, b]$.
3. If $f'(x) = 0$ for all $x \in (a, b)$, then $f(x)$ is constant on $[a, b]$.

Determining Intervals of Monotonicity

To find intervals where a function increases or decreases:

1. Find $f'(x)$.
2. Set $f'(x) = 0$ and find all real roots (stationary points) as well as points where $f'(x)$ is undefined. These partition the real line into disjoint test intervals.
3. Test the algebraic sign of $f'(x)$ in each interval: intervals where $f'(x) > 0$ are intervals of increase; intervals where $f'(x) < 0$ are intervals of decrease.

---

4. Critical Points & Stationary Points

Definitions

• A critical point $x = c$ is a point in the domain of $f$ where either:
  1. $f'(c) = 0$, OR
  2. $f'(c)$ is undefined (does not exist).
• A stationary point is specifically a critical point where the tangent is horizontal: $f'(c) = 0$.

ECAT Key Distinction: All stationary points are critical points, but not all critical points are stationary points! For instance, $f(x) = |x|$ has a critical point at $x = 0$ because $f'(0)$ does not exist, but it is not a stationary point.

Fermat's Theorem on Local Extrema

If $f(x)$ has a local extremum (local maximum or local minimum) at an interior point $c$ of its domain, and if $f'(c)$ exists, then:

$f'(c) = 0$

Caution: The converse of Fermat's Theorem is false. A stationary point is not necessarily a local extremum. For example, for $f(x) = x^3$, $f'(0) = 0$, but $x = 0$ is neither a local maximum nor a local minimum; it is a horizontal point of inflection.

---

5. Tests for Local Extrema

1. The First Derivative Test

Let $c$ be a critical point of a continuous function $f(x)$:

• Local Maximum: If $f'(x)$ changes sign from positive to negative as $x$ increases through $c$ (i.e., $f'(x) > 0$ for $x < c$ and $f'(x) < 0$ for $x > c$), then $f(c)$ is a local maximum.
• Local Minimum: If $f'(x)$ changes sign from negative to positive as $x$ increases through $c$ (i.e., $f'(x) < 0$ for $x < c$ and $f'(x) > 0$ for $x > c$), then $f(c)$ is a local minimum.
• No Local Extremum: If $f'(x)$ does not change sign as $x$ passes through $c$ (it remains positive on both sides or negative on both sides), then $c$ is a point of inflection, not an extremum.

2. The Second Derivative Test

Let $f$ be twice differentiable at a stationary point where $f'(c) = 0$:

1. If $f''(c) < 0$, the graph is concave down at $c$, and $f(c)$ is a local maximum.
2. If $f''(c) > 0$, the graph is concave up at $c$, and $f(c)$ is a local minimum.
3. If $f''(c) = 0$, the second derivative test is inconclusive! The point could be a local maximum, local minimum, or a point of inflection. You must revert to the First Derivative Test or use the higher-order derivative test.

Higher-Order Derivative Test

If $f'(c) = f''(c) = \dots = f^{(n-1)}(c) = 0$ and $f^{(n)}(c) \neq 0$:

• If $n$ is odd, $f(c)$ is not an extremum (it is a point of inflection).
• If $n$ is even and $f^{(n)}(c) < 0$, $f(c)$ is a local maximum.
• If $n$ is even and $f^{(n)}(c) > 0$, $f(c)$ is a local minimum.

---

6. Concavity and Points of Inflection

Concavity

Let $f(x)$ be twice differentiable on an interval $I$:

• The curve is concave upward (convex downward, shaped like $\cup$) on $I$ if $f''(x) > 0$ for all $x \in I$. All tangents lie below the curve.
• The curve is concave downward (convex upward, shaped like $\cap$) on $I$ if $f''(x) < 0$ for all $x \in I$. All tangents lie above the curve.

Point of Inflection

A point $(c, f(c))$ on a continuous curve where the concavity changes from upward to downward or downward to upward is called a point of inflection.

• Necessary Condition: If $(c, f(c))$ is a point of inflection and $f''(c)$ exists, then $f''(c) = 0$.
• Sufficient Condition: $f''(c) = 0$ (or undefined) AND $f''(x)$ strictly changes algebraic sign across $x = c$.

---

7. Real-World Optimization Problems

Applied optimization questions on the ECAT require modeling a physical scenario, establishing a single-variable objective function, and determining its global extrema under geometric constraints.

Standard Optimization Protocol

1. Identify Variables: Assign symbols to the quantity to be optimized (e.g., area $A$, volume $V$, cost $C$) and independent dimensions ($x, y, r, h$).
2. Formulate the Constraint: Use problem geometry to write an equation relating variables (e.g., fixed perimeter $2x + 2y = P$, fixed surface area $S$).
3. Express Objective Function: Substitute the constraint into the target function to express it in terms of a single variable: $A = f(x)$.
4. Differentiate & Solve: Find $f'(x) = 0$ to identify stationary points.
5. Confirm Extremum: Verify via $f''(x) < 0$ (for maximum) or $f''(x) > 0$ (for minimum).

Classical ECAT Optimization Shortcuts

• Maximum Area of Rectangle with Fixed Perimeter $P$: A rectangle with fixed perimeter $P = 2x + 2y$ achieves maximum area when it is a square: $x = y = \frac{P}{4}$, with $A_{\max} = \left(\frac{P}{4}\right)^2$.
• Three-Sided Enclosure (Against a Wall): If fencing of length $L$ encloses three sides ($x + 2y = L$), the maximum area occurs when the parallel side equals twice the perpendicular side: $x = \frac{L}{2}$ and $y = \frac{L}{4}$, yielding $A_{\max} = \frac{L^2}{8}$.
• Open-Top Box from Square Sheet: From a square sheet of side length $a$, cutting corner squares of side $x$ and folding upward creates an open box of volume $V(x) = x(a - 2x)^2$. Maximum volume occurs when $x = \frac{a}{6}$, giving $V_{\max} = \frac{2a^3}{27}$.