Understanding Max. Value and Min. Value of a Function Using Limits

When studying functions in calculus, one of the key concepts is determining the maximum and minimum values that a function can take. These values are important in various applications such as optimization problems, curve sketching, and understanding the behavior of functions. Limits play a crucial role in finding these values, especially when dealing with continuous functions.

Definitions

Before diving into the methods of finding maxima and minima using limits, let's define some terms:

• Maximum Value: The highest point on the graph of a function within a given domain.
• Minimum Value: The lowest point on the graph of a function within a given domain.
• Local Maximum: The highest point in a small, localized region of the graph.
• Local Minimum: The lowest point in a small, localized region of the graph.
• Global (Absolute) Maximum: The highest point on the entire graph of the function.
• Global (Absolute) Minimum: The lowest point on the entire graph of the function.
• Limit: The value that a function approaches as the input approaches some value.

Table of Differences

Aspect	Maximum Value	Minimum Value
Definition	Highest value a function can take	Lowest value a function can take
Local vs Global	Local max or global max	Local min or global min
Symbol	Often denoted by $f_{\text{max}}$	Often denoted by $f_{\text{min}}$
Limit Behavior	$\lim_{x \to c^-} f(x) \leq f(c)$ and $\lim_{x \to c^+} f(x) \leq f(c)$ for local max	$\lim_{x \to c^-} f(x) \geq f(c)$ and $\lim_{x \to c^+} f(x) \geq f(c)$ for local min
Critical Points	Derivative equals zero or undefined	Derivative equals zero or undefined
Second Derivative Test	$f''(c) < 0$ for a local max	$f''(c) > 0$ for a local min

Formulas

To find the maxima and minima of a function using limits, we often rely on the first and second derivative tests:

• First Derivative Test: If $f'(c) = 0$ and $f'(x)$ changes sign from positive to negative at $c$, then $f(c)$ is a local maximum. If $f'(x)$ changes sign from negative to positive at $c$, then $f(c)$ is a local minimum.

• Second Derivative Test: If $f'(c) = 0$ and $f''(c) > 0$, then $f(c)$ is a local minimum. If $f''(c) < 0$, then $f(c)$ is a local maximum.

Examples

Example 1: Using the First Derivative Test

Consider the function $f(x) = x^2 - 4x + 3$. To find its maxima and minima:

1. Find the derivative: $f'(x) = 2x - 4$.
2. Set the derivative equal to zero: $2x - 4 = 0 \Rightarrow x = 2$.
3. Analyze the sign of $f'(x)$ around $x = 2$:
   • For $x < 2$, $f'(x) > 0$ (positive).
   • For $x > 2$, $f'(x) < 0$ (negative).
4. Since $f'(x)$ changes from positive to negative at $x = 2$, $f(2)$ is a local maximum.

Example 2: Using the Second Derivative Test

Consider the same function $f(x) = x^2 - 4x + 3$.

1. Find the second derivative: $f''(x) = 2$.
2. Since $f''(x) > 0$ for all $x$, any critical point is a local minimum.
3. However, since we found $x = 2$ to be a critical point and $f''(x)$ is always positive, this indicates that $x = 2$ is actually a point of inflection, and there are no local maxima or minima for this function.

Example 3: Using Limits to Find Global Extrema

Consider the function $f(x) = \frac{1}{x^2}$ on the interval $[1, \infty)$.

1. Find the limit as $x$ approaches infinity: $\lim_{x \to \infty} \frac{1}{x^2} = 0$.
2. Since $f(x)$ is always decreasing and approaches $0$ as $x$ goes to infinity, the global minimum is at $x = \infty$ and $f_{\text{min}} = 0$.
3. The global maximum is at the left endpoint of the interval, $x = 1$, where $f(1) = 1$.

By understanding these concepts and applying the appropriate tests and limit calculations, one can determine the maximum and minimum values of a function, which is essential for solving a wide range of mathematical problems.