Let f(x)=x^2+p x+q . Find the values of p and q such that f(1)=3 is an extreme value of f on [0,

step 1 In the version, we have given with the function, which is x squared plus px plus q, and we have

Let f(x)=x^2+p x+q . Find the values of p and q such that...

Key Concepts

Parameter Determination in Functions

Parameter determination in functions involves finding unknown constants by using given conditions or constraints. In problems where certain function values or behaviors (such as an extreme value occurring at a specific point) are provided, these conditions can be substituted into the function to form equations that allow for solving the parameters. This method is widely applicable in algebra and applied mathematics to tailor function behavior to meet specific criteria.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, generally expressed in the form f(x) = ax² + bx + c, with a ? 0. They have a parabolic graph that opens upward when a > 0 (implying a minimum at the vertex) and downward when a < 0 (implying a maximum). Understanding the general behavior and shape of quadratic functions is fundamental in analyzing their extreme values and overall properties.

Vertex of a Parabola

The vertex of a parabola is the point where the function attains its minimum or maximum value, depending on the direction of its opening. For a quadratic function, the vertex can be found either by completing the square or using the formula x = -b/(2a). Recognizing the role of the vertex is essential in optimization problems, as it provides a systematic way to identify the candidate for an extreme value within a given domain.

Extreme Value Theorem

The Extreme Value Theorem states that a continuous function defined on a closed and bounded interval will always attain both a maximum and minimum value. This theorem is particularly important in optimization as it implies that when considering a function on a closed interval, one must examine the function's values at both the critical points and the endpoints of the interval.

Critical Points and Derivatives

Critical points occur where the derivative of a function is zero or undefined and are potential locations for local extrema. For differentiable functions like quadratics, setting the derivative equal to zero identifies the vertex, which is the candidate for a local extreme. This concept is a core element of calculus-based optimization and analysis of functions.

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