An extremum, in the context of mathematics, refers to a point at which a function reaches either a maximum or a minimum value. It is a critical point where the function's derivative is equal to zero or undefined, and the function's value is greater than or less than the values in the immediate vicinity.
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Quadratic functions, which are of the form $f(x) = ax^2 + bx + c$, always have at most one local extremum, which is either a maximum or a minimum.
The x-coordinate of the extremum of a quadratic function can be found by using the formula $x = -b/(2a)$, where $a$ and $b$ are the coefficients of the function.
The y-coordinate of the extremum of a quadratic function can be found by substituting the x-coordinate into the original function, $f(x)$.
The type of extremum (maximum or minimum) can be determined by the sign of the coefficient $a$: if $a > 0$, the function has a minimum; if $a < 0$, the function has a maximum.
Quadratic functions are symmetric about their axis of symmetry, which passes through the extremum.
Review Questions
Explain how to identify the extremum of a quadratic function and describe its key features.
To identify the extremum of a quadratic function $f(x) = ax^2 + bx + c$, first find the x-coordinate of the extremum using the formula $x = -b/(2a)$. Then, substitute this x-value into the original function to find the y-coordinate of the extremum. The type of extremum (maximum or minimum) is determined by the sign of the coefficient $a$: if $a > 0$, the function has a minimum; if $a < 0$, the function has a maximum. Quadratic functions are symmetric about their axis of symmetry, which passes through the extremum.
Distinguish between a local extremum and a global extremum, and explain how they differ in the context of quadratic functions.
A local extremum is a point where a function attains a maximum or minimum value within a small neighborhood around that point. A global extremum, on the other hand, is a point where a function attains the absolute maximum or minimum value over the entire domain of the function. For quadratic functions, which are of the form $f(x) = ax^2 + bx + c$, there is at most one local extremum, which is either a maximum or a minimum. This local extremum is also the global extremum, as quadratic functions are symmetric and have at most one critical point, which corresponds to the extremum.
Analyze how the coefficient $a$ in the quadratic function $f(x) = ax^2 + bx + c$ affects the type and location of the extremum.
The coefficient $a$ in the quadratic function $f(x) = ax^2 + bx + c$ plays a crucial role in determining the type and location of the extremum. If $a > 0$, the function has a minimum extremum, and if $a < 0$, the function has a maximum extremum. The x-coordinate of the extremum is given by the formula $x = -b/(2a)$, which means that the location of the extremum is directly influenced by the values of $a$ and $b$. Additionally, the sign of $a$ determines the concavity of the parabolic graph, with $a > 0$ indicating an upward-facing parabola and $a < 0$ indicating a downward-facing parabola.
Related terms
Local Extremum: A local extremum is a point where a function attains a maximum or minimum value within a small neighborhood around that point.
Global Extremum: A global extremum is a point where a function attains the absolute maximum or minimum value over the entire domain of the function.
Critical Point: A critical point is a point where the derivative of a function is equal to zero or undefined, indicating a potential extremum.