Monotonicity Based Problems - Inequality Based Using First Derivative

Monotonicity in mathematics refers to the behavior of a function as either non-decreasing or non-increasing within a given interval. Understanding monotonicity is crucial when solving inequality-based problems, as it helps to determine the intervals where a function is above or below a certain value. The first derivative of a function is a powerful tool in analyzing monotonicity.

Understanding the First Derivative

The first derivative of a function, denoted as $f'(x)$, gives us the slope of the tangent line to the function at any point $x$. It tells us how the function is changing at that point:

• If $f'(x) > 0$, the function is increasing at $x$.
• If $f'(x) < 0$, the function is decreasing at $x$.
• If $f'(x) = 0$, the function may have a local maximum, minimum, or a point of inflection at $x$.

Using the First Derivative to Determine Monotonicity

To use the first derivative to determine where a function is monotonic, follow these steps:

1. Find the first derivative of the function, $f'(x)$.
2. Solve the inequality $f'(x) > 0$ to find where the function is increasing.
3. Solve the inequality $f'(x) < 0$ to find where the function is decreasing.
4. Identify any critical points where $f'(x) = 0$ and analyze the behavior of the function around these points.

Table of Monotonicity and First Derivative

Condition on $f'(x)$	Monotonicity of $f(x)$	Interpretation
$f'(x) > 0$	Increasing	Function moves upwards as $x$ increases
$f'(x) < 0$	Decreasing	Function moves downwards as $x$ increases
$f'(x) = 0$	Stationary	Potential local maxima, minima, or inflection points

Examples

Example 1: Determine Monotonicity

Let's find where the function $f(x) = x^3 - 3x^2 + 4$ is increasing or decreasing.

1. Find the first derivative: $f'(x) = 3x^2 - 6x$.
2. Set the first derivative greater than zero to find where $f(x)$ is increasing: $$3x^2 - 6x > 0$$ $$x(3x - 6) > 0$$ The solutions are $x < 0$ or $x > 2$.
3. Set the first derivative less than zero to find where $f(x)$ is decreasing: $$3x^2 - 6x < 0$$ $$x(3x - 6) < 0$$ The solution is $0 < x < 2$.

Therefore, $f(x)$ is increasing on $(-\infty, 0) \cup (2, \infty)$ and decreasing on $(0, 2)$.

Example 2: Solving an Inequality Using Monotonicity

Solve the inequality $x^3 - 3x^2 + 4 < 5$ using monotonicity.

1. Rewrite the inequality: $x^3 - 3x^2 - 1 < 0$.
2. Find the first derivative: $f'(x) = 3x^2 - 6x$.
3. Determine the critical points by setting $f'(x) = 0$: $$3x^2 - 6x = 0$$ $$x(3x - 6) = 0$$ The critical points are $x = 0$ and $x = 2$.
4. Test intervals around the critical points to determine where $f(x) < 0$:
   • For $x < 0$, choose $x = -1$: $f(-1) = -1 - 3 - 1 < 0$ (satisfies the inequality).
   • For $0 < x < 2$, choose $x = 1$: $f(1) = 1 - 3 - 1 < 0$ (satisfies the inequality).
   • For $x > 2$, choose $x = 3$: $f(3) = 27 - 27 - 1 < 0$ (does not satisfy the inequality).

The solution to the inequality is $x \in (-\infty, 2)$.

By understanding the relationship between the first derivative and monotonicity, we can effectively solve inequality-based problems and determine the behavior of functions within specific intervals.