The Vertex of a Parabola Occurs Where the Derivative of Its

question 218

Multiple Choice

The vertex of a parabola occurs where the derivative of its equation is 0, so an equation associated with the equation of a
parabola can be solved to find the x-coordinate of the vertex. Find the vertex of the parabola, given the equation of the
parabola and its associated equation.
-Between $x = 0$ and $x = 8$ , the largest value of the expression $y = x ^ { 3 } - 15 x ^ { 2 } + 63 x + 18$ occurs where $3 x ^ { 2 } - 30 x + 63 = 0$ . Of the values of $x$ that satisfy $3 x ^ { 2 } - 30 x + 63 = 0$ , which gives the maximum value of $y = x ^ { 3 } - 15 x ^ { 2 } + 63 x + 18 ?$

• A) \(x = 2\)
• B) \(x = 3\)
• C) \(x = 6\)
• D) \(x = 7\)

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Definitions:

Substitution Effect

The change in consumption patterns due to a change in relative prices, leading consumers to substitute one good for another.

Wage Increase

An upward adjustment in the salary or earnings paid to an employee.

Utility Maximizing

Acting in a way to achieve the highest level of satisfaction possible with available resources.

Demand For Labor

The total amount of labor that employers want to hire at various wage rates, contingent on factors such as technology, the price of the goods being produced, and the availability of other inputs.