Pharmaceuticals When a certain drug is taken orally, the...

Pharmaceuticals When a certain drug is taken orally, the concentration of the drug in the patient's bloodstream after $t$ minutes is given by $C(t)=0.06 t-0.0002 t^{2},$ where $0 \leq t \leq 240$ and the concentration is measured in mg/$L$. When is the maximum serum concentration reached, and what is that maximum concentration?

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Key Concepts

Quadratic Functions

Quadratic functions are algebraic expressions of the form f(x) = ax² + bx + c, where a, b, and c are constants and a ? 0. Their graphs are parabolas, which can open upward or downward. The nature of the parabola, including its symmetry and curvature, makes it a fundamental tool for modeling situations where there is a single peak or trough, such as determining maximum or minimum values.

Vertex of a Parabola

The vertex of a parabola is the point where the function attains its maximum or minimum value, depending on whether the parabola opens downward or upward. This is a key concept in optimization because, for a downward-opening parabola, the vertex represents the highest point, and for an upward-opening parabola, it represents the lowest point. The vertex can be found using the formula x = -b/(2a) when the quadratic function is in the standard form.

Optimization Using Critical Point Analysis

Optimization in continuous functions involves finding the highest or lowest values that a function can take within a certain domain. For quadratic functions, this process is streamlined by computing the vertex or employing derivative techniques to identify points where the rate of change is zero. This process is essential for solving problems in various fields where identifying optimal conditions is required.

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