key term - Extrema

Definition

Extrema, in the context of mathematical functions, refer to the points where a function attains its maximum or minimum values. These critical points, known as local maxima and local minima, are important in understanding the behavior and properties of various functions, including quadratic functions and trigonometric functions like sine and cosine.

5 Must Know Facts For Your Next Test

1. For quadratic functions, the extrema (maximum or minimum) occur at the vertex of the parabola, which can be found by using the formula $x = -b/2a$, where $a$ and $b$ are the coefficients of the quadratic function.
2. The sign of the leading coefficient $a$ in a quadratic function determines whether the extremum is a maximum (if $a < 0$) or a minimum (if $a > 0$).
3. For the graphs of sine and cosine functions, the extrema occur at the points where the derivative of the function is zero, which correspond to the maximum and minimum values of the function.
4. The amplitude of a sine or cosine function determines the vertical distance between the extrema, while the period determines the horizontal distance between the extrema.
5. Extrema play a crucial role in optimization problems, where the goal is to find the maximum or minimum value of a function subject to certain constraints.

Review Questions

• Explain how the leading coefficient $a$ in a quadratic function $f(x) = ax^2 + bx + c$ determines the type of extremum (maximum or minimum) of the function.
  • The sign of the leading coefficient $a$ in a quadratic function $f(x) = ax^2 + bx + c$ determines the type of extremum. If $a < 0$, the function has a maximum at the vertex of the parabola. If $a > 0$, the function has a minimum at the vertex of the parabola. This is because the sign of $a$ determines the concavity of the parabola, with $a < 0$ resulting in a downward-facing parabola (maximum) and $a > 0$ resulting in an upward-facing parabola (minimum).
• Describe how the extrema of sine and cosine functions are related to the derivatives of these functions.
  • The extrema of sine and cosine functions occur at the points where the derivative of the function is equal to zero. This is because the derivative of a function represents the rate of change of the function, and at the extrema, the rate of change is zero, indicating a maximum or minimum value. For sine and cosine functions, the extrema correspond to the points where the function values are at their maximum or minimum, and the derivative is equal to zero at these critical points.
• Explain how extrema are used in optimization problems and discuss their importance in finding the optimal solution.
  • Extrema play a crucial role in optimization problems, where the goal is to find the maximum or minimum value of a function subject to certain constraints. By identifying the extrema of the function, we can determine the points where the function attains its maximum or minimum values, which are the potential solutions to the optimization problem. The ability to find the extrema of a function is essential in solving a wide range of optimization problems, from finding the maximum profit for a business to determining the optimal design of an engineering system. Extrema provide valuable information about the behavior of the function and help us make informed decisions to find the optimal solution.

Related terms

Local Maximum:

A point on a function's graph where the function value is greater than the function values at all nearby points.

Local Minimum:

A point on a function's graph where the function value is less than the function values at all nearby points.

Critical Point:

A point on a function's graph where the derivative of the function is equal to zero or undefined, indicating a potential extremum.