In Exercises 89–92, find the values of b such that the function has the given maximum or minimum

step 1 So I have f of x equals x squared plus b times x minus 25, and the minimal value is minus 50. So

In Exercises 89–92, find the values of b such that the...

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Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial expressed as f(x) = ax² + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0, which directly influences whether the function has a minimum or a maximum value.

Vertex Form of a Quadratic

The vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form simplifies the process of identifying the function's extreme value since the vertex yields either the maximum or minimum point of the parabola.

Completing the Square

Completing the square is a method used to convert a quadratic expression from its standard form into vertex form. By rewriting the quadratic expression as a perfect square plus a constant, one can easily identify the vertex of the parabola, which is essential in solving optimization problems.

Optimization through the Vertex

Optimization in the context of quadratic functions involves finding the minimum or maximum value, which occurs at the vertex. Once the quadratic is expressed in vertex form, the coordinates of the vertex directly provide the value of the function's extreme point, enabling one to adjust parameters to achieve a specified maximum or minimum.

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