Finding the Maximum: The Peak of a Graph
In mathematics, a maximum is a special point on a graph where the y-coordinate reaches a peak value compared to its neighboring points. It is a type of turning point or vertex that represents the highest output value of a function in a specific interval. Identifying maxima is crucial for solving real-world optimization problems, such as finding the greatest profit, the highest point of a ball's flight, or the most efficient dimensions for a container.
The Visual Nature of a Maximum
Imagine you are climbing a hill. The very top of the hill is the highest point you reach before the path starts going down. On a graph, this "top of the hill" is called a maximum. More precisely, a maximum is a point where the $y$-value is greater than the $y$-values of the points immediately to its left and right. This means it's a local peak. It's a turning point because the graph changes direction at that spot—it goes from increasing (upward slope) to decreasing (downward slope).
For a simple example, consider the graph of the quadratic function $y = -x^2 + 4$. This graph is a downward-opening parabola. Its vertex, or highest point, is at $(0, 4)$. Check the points around it: $(-1, 3)$ and $(1, 3)$. The $y$-value $4$ is indeed greater than $3$, confirming it as a maximum.
Different Types of Maximums
Not all maximums are created equal. It's important to distinguish between two main types, as this helps us understand the overall behavior of a function.
| Type | Definition | Also Called | Visual Clue |
|---|---|---|---|
| Local (or Relative) Maximum | A point whose $y$-coordinate is greater than that of all nearby points. It is the highest point in its immediate neighborhood. | Relative Max | The top of any individual hill on the graph. A graph can have multiple local maximums. |
| Global (or Absolute) Maximum | A point whose $y$-coordinate is greater than or equal to that of all other points on the entire graph (within the domain[1] being considered). | Absolute Max | The highest point on the entire graph. This is the tallest peak among all hills. |
A helpful analogy is a mountain range. Each mountain summit is a local maximum. The summit of the tallest mountain in the entire range is the global maximum. A global maximum is always a local maximum, but a local maximum is not necessarily the global one.
How to Find a Maximum: A Step-by-Step Guide
Finding maximums can be done by simple observation for basic graphs, but for more complex functions, we need a reliable method. Here is a progression of techniques from elementary to high school level.
1. Look at the graph. Identify all the "hilltops."
2. For each hilltop, check that the $y$-value is higher than the points to its left and right.
3. The hilltop with the highest $y$-value of all is the global maximum.
For a parabola given by $y = ax^2 + bx + c$:
• If $a < 0$, the parabola opens downward and has a maximum at its vertex.
• The $x$-coordinate of the vertex is $x = -\frac{b}{2a}$.
• Substitute this $x$-value back into the equation to find the maximum $y$-value.
Example: Find the maximum of $y = -2x^2 + 8x + 1$.
Since $a = -2$ (negative), the parabola opens downward and has a maximum.
The $x$-coordinate of the vertex is: $x = -\frac{8}{2(-2)} = -\frac{8}{-4} = 2$.
Now find $y$ when $x=2$: $y = -2(2)^2 + 8(2) + 1 = -8 + 16 + 1 = 9$.
Therefore, the maximum point is $(2, 9)$.
For smoother curves, calculus gives a powerful tool. At a maximum (or minimum), the slope of the tangent line is zero.
1. Find the derivative[2], $f'(x)$, which gives the slope.
2. Set $f'(x) = 0$ and solve for $x$. These are called critical points.
3. Test these points: If the derivative changes from positive (increasing) to negative (decreasing) at a critical point, that point is a local maximum.
From Theory to Practice: Real-World Applications
The concept of a maximum is not just a mathematical abstraction; it is the key to optimization—finding the best possible outcome under given conditions. Here are concrete examples across different fields.
1. Business & Economics (Maximizing Profit): A company's profit $P$ often depends on the number of items sold, $x$. The relationship can be modeled by a function like $P(x) = -5x^2 + 300x - 2000$. The graph of this profit function is a downward-opening parabola. The vertex of this parabola represents the number of items to sell $(x)$ to achieve the maximum possible profit and the value of that profit $(P)$. Finding this maximum is critical for business planning.
2. Physics (Projectile Motion): When you throw a ball straight up, its height $h$ (in meters) after $t$ seconds can be given by $h(t) = -4.9t^2 + 20t + 1.5$. This is again a quadratic function. The maximum point on this graph tells you the peak height the ball reaches and the time at which it reaches that height. This is a direct application of finding the vertex.
3. Everyday Decisions: Imagine you have a fixed length of fence to create a rectangular vegetable garden against your house. You want to enclose the largest possible area. If the width of the garden is $x$ feet, the area $A$ can be expressed as a quadratic function in terms of $x$: $A = x \times (\text{length})$. The maximum of this area function tells you the ideal width to use for the biggest garden.
Important Questions
A: Typically, no. For a point to be a smooth local maximum, the graph needs to be relatively smooth and continuous[3] around it. At a sharp corner, the idea of "points immediately to its left and right" can be ambiguous because the slope changes abruptly. While the highest point at a corner might be considered a maximum in a broad sense, the standard methods for finding maxima (like setting the derivative to zero) do not work there because the derivative does not exist at a sharp corner.
A: They are both turning points (vertices), but they represent opposite extremes. A maximum is a peak—the $y$-coordinate is greater than neighboring points. A minimum is a valley—the $y$-coordinate is less than neighboring points. On a parabola, if it opens upward, the vertex is a minimum; if it opens downward, the vertex is a maximum.
A: Not necessarily. A turning point is where a graph changes direction. While most common turning points are maxima (change from increasing to decreasing) or minima (change from decreasing to increasing), some more complex functions can have points where the graph flattens out (like in $y = x^3$ at $x=0$) without being a peak or a valley. This is called a point of inflection[4].
Understanding the concept of a maximum—a turning point where the $y$-value is greater than its immediate neighbors—is a cornerstone of mathematical thinking. From the simple visual identification of a hilltop on a graph to the algebraic calculation of a parabola's vertex and the calculus-based derivative test, the journey to find maxima equips us with powerful problem-solving tools. This knowledge transcends pure math, enabling us to model and optimize real-world scenarios in business, science, and daily life. By mastering how to find and interpret maximums, we learn to identify optimal outcomes and understand the peak behaviors of changing systems.
Footnote
[1] Domain: The set of all possible input values (usually $x$-values) for a function.
[2] Derivative: A concept from calculus that measures the instantaneous rate of change of a function. It represents the slope of the tangent line at any point on the graph.
[3] Continuous: A function is continuous if its graph can be drawn without lifting the pencil from the paper. It has no breaks, jumps, or holes.
[4] Point of Inflection: A point on a curve where the curvature changes sign, i.e., the graph changes from being concave up (like a cup) to concave down (like a cap), or vice-versa.