Application of Derivatives Class 12 Notes

Understanding Monotonicity & Rates of Change

The derivative of a function is not merely a formulaic tool; geometrically, it represents the slope of the tangent to the curve at any point. Physically, it measures the instantaneous rate of change of one quantity with respect to another.

1. Rate of Change of Quantities

If a variable $y$ varies with another variable $x$ such that $y = f(x)$, then the derivative $\frac{dy}{dx}$ represents the rate of change of $y$ with respect to $x$. If both $x$ and $y$ vary with respect to time $t$, we apply the chain rule:
$\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$

2. Monotonicity (Increasing and Decreasing Functions)

Monotonicity describes the direction of a function's curve over an interval $I$:

• Strictly Increasing: As $x$ increases, $f(x)$ strictly increases. Its derivative is positive: $f'(x) > 0$ for all $x \in (a, b)$.
• Strictly Decreasing: As $x$ increases, $f(x)$ strictly decreases. Its derivative is negative: $f'(x) < 0$ for all $x \in (a, b)$.
• Stationary Points: Points where the tangent to the curve is perfectly horizontal, implying $f'(x) = 0$.

Essential Exam Terminology

Rate Measurer

The derivative $\frac{dy}{dx}$ which determines how fast quantity $y$ changes relative to a unit change in quantity $x$.

Increasing Function

A function $f(x)$ is increasing in an interval $I$ if $x_1 < x_2$ implies $f(x_1) \le f(x_2)$ for all $x_1, x_2 \in I$.

Strictly Increasing Function

A function $f(x)$ is strictly increasing in an interval $I$ if $x_1 < x_2$ implies $f(x_1) < f(x_2)$ for all $x_1, x_2 \in I$.

Critical Point

A point $c$ in the domain of $f(x)$ where either $f'(c) = 0$ or $f(x)$ is not differentiable.

Point of Inflection

A point on a curve where the second derivative $f''(x)$ changes sign, indicating a change in the direction of concavity.

Local Extremum

A collective term for local maximum or local minimum points on a function's curve within a specific neighborhood.

Must Remember: Key Formulas & Derivative Tests

• Rate of change formula: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, provided $\frac{dx}{dt} \neq 0$.
• A function $f(x)$ is strictly increasing in $[a, b]$ if $f'(x) > 0$ for each $x \in (a, b)$.
• A function $f(x)$ is strictly decreasing in $[a, b]$ if $f'(x) < 0$ for each $x \in (a, b)$.
• Critical points are the fundamental candidates for local maxima and local minima.
• First Derivative Test: If $f'(x)$ changes sign from positive to negative as $x$ increases through $c$, then $c$ is a point of local maximum.
• First Derivative Test: If $f'(x)$ changes sign from negative to positive as $x$ increases through $c$, then $c$ is a point of local minimum.
• Second Derivative Test: If $f'(c) = 0$ and $f''(c) < 0$, then $x = c$ is a point of local maximum.
• Second Derivative Test: If $f'(c) = 0$ and $f''(c) > 0$, then $x = c$ is a point of local minimum.
• To find the Absolute Maxima/Minima on a closed interval $[a, b]$, always evaluate and compare $f(x)$ at all critical points AND the boundary points $a$ and $b$.

Step-by-Step: Finding Local Extrema Using the Second Derivative Test

1. Find the First Derivative — Compute $f'(x)$ for the given function $f(x)$.
2. Identify Critical Points — Solve the algebraic equation $f'(x) = 0$ to extract the critical values, say $x = c_1, c_2, \dots$
3. Find the Second Derivative — Compute the second-order derivative $f''(x)$ by differentiating $f'(x)$.
4. Evaluate at Critical Points — Substitute each critical value $c$ into $f''(x)$ to find $f''(c)$.
5. Analyze and Conclude — If $f''(c) > 0$, then $c$ is a local minimum. If $f''(c) < 0$, then $c$ is a local maximum. If $f''(c) = 0$, the test fails; immediately fall back on the First Derivative Test.

Monotonic Function Types Comparison

Worked Revision Examples

• Example 1: Find the intervals in which the function $f(x) = x^2 - 4x + 6$ is (a) strictly increasing, (b) strictly decreasing. 1. Differentiate $f(x)$: $f'(x) = 2x - 4$. 2. Set $f'(x) = 0 \implies 2x - 4 = 0 \implies x = 2$. 3. The point $x = 2$ divides the real line into two intervals: $(-\infty, 2)$ and $(2, \infty)$. For $x \in (-\infty, 2)$, pick $x = 0 \implies f'(0) = -4 < 0$. Thus, $f(x)$ is strictly decreasing in $(-\infty, 2]$. For $x \in (2, \infty)$, pick $x = 3 \implies f'(3) = 2 > 0$. Thus, $f(x)$ is strictly increasing in $[2, \infty)$.
• Example 2: Find the absolute maximum and minimum values of $f(x) = 3x^4 - 8x^3 + 12x^2 - 48x + 25$ on the closed interval $[0, 3]$. 1. Find $f'(x)$: $f'(x) = 12x^3 - 24x^2 + 24x - 48 = 12(x^3 - 2x^2 + 2x - 4) = 12(x^2(x - 2) + 2(x - 2)) = 12(x^2 + 2)(x - 2)$. 2. Set $f'(x) = 0$. Since $x^2 + 2 \neq 0$, the only real critical point is $x = 2$. 3. Evaluate $f(x)$ at critical point and endpoints: $f(0) = 25$ $f(2) = 3(16) - 8(8) + 12(4) - 48(2) + 25 = 48 - 64 + 48 - 96 + 25 = -39$ * $f(3) = 3(81) - 8(27) + 12(9) - 48(3) + 25 = 243 - 216 + 108 - 144 + 25 = 16$ 4. Conclusion: Absolute Maximum Value is $25$ at $x = 0$. Absolute Minimum Value is $-39$ at $x = 2$.

Board Exam Trap Alerts

• Absolute Max/Min Trap: When the question asks for "Absolute" or "Global" extrema on a closed interval $[a, b]$, do not use the Second Derivative Test. Instead, directly compute $f(x)$ at all critical points AND the boundary points $a$ and $b$, then pick the largest and smallest values.
• Strict vs. Non-Strict Inequalities: Ensure you differentiate between "strictly increasing" ($f'(x) > 0$ on open interval, closed brackets on final answer if continuous) and standard "increasing" ($f'(x) \ge 0$).
• Failing of 2nd Derivative Test: If $f''(c) = 0$, the test is inconclusive. It does not necessarily mean $c$ is a point of inflection. You must revert to the First Derivative Test to check for local extrema.

Quick Revision Check

• The total area of a circle is expanding. What is the rate of change of the area $A$ with respect to its radius $r$ when $r = 7$ cm? $A = \pi r^2 \implies \frac{dA}{dr} = 2\pi r$. Substituting $r = 7$, we get $\frac{dA}{dr} = 14\pi \text{ cm}^2/\text{cm}$.
• Prove that $f(x) = 3x + 17$ is strictly increasing on $\mathbb{R}$. Differentiating, $f'(x) = 3$. Since $3 > 0$ for all real numbers $x$, $f(x)$ is strictly increasing on $\mathbb{R}$.
• Identify the critical points of the function $f(x) = x^3 - 3x + 3$. $f'(x) = 3x^2 - 3$. Setting $f'(x) = 0 \implies 3(x^2 - 1) = 0 \implies x = 1$ and $x = -1$. The critical points are $x = \pm 1$.
• If $f'(x) > 0$ for all $x \in (a, b)$ except at a finite number of points where it is zero, is the function still strictly increasing? Yes, if the points where $f'(x) = 0$ do not form an interval (i.e., they are isolated points), the function is still considered strictly increasing.

Frequently Asked Questions