Advertising. Brody Electronics estimates that it will sell N...

Advertising. Brody Electronics estimates that it will sell N units of a new toy after spending $a$ thousands of dollars on advertising, where $N(a)=-a^{2}+300 a+6, \quad 0 \leq a \leq 300$
Find the relative extrema and sketch a graph of the function.

Key Concepts

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. For quadratic functions, the vertex can be computed using the formula derived from the standard form and is a critical component in graphing and optimization tasks.

Graphing Quadratic Functions

Graphing quadratic functions involves plotting a parabola, which is the characteristic graph of a quadratic function. Key features include the vertex, axis of symmetry, and intercepts. Understanding these features is essential for interpreting the behavior of the function and for sketching an accurate graph.

Constrained Domain Optimization

When a function is defined on a closed interval, the overall maximum or minimum values may occur at the critical points or at the endpoints of the interval. This concept, known as constrained domain optimization, ensures that all possible candidates are checked in order to determine the global relative extrema.

Derivative and Critical Points

The derivative of a function represents its instantaneous rate of change, and by setting the derivative equal to zero, one can identify critical points. These are the candidates for relative maxima or minima because they indicate where the slope of the function is zero, which is necessary for the existence of local extrema.

Relative Extrema

Relative extrema refer to the maximum or minimum points of a function within a certain interval. These are the specific points where the function changes direction from increasing to decreasing or vice versa, and they are crucial in determining the optimal values that a function can attain in a given context.

Second Derivative Test

The second derivative test is a method used to classify the critical points found by the first derivative. It examines the concavity of the function; if the second derivative is negative at a critical point, the function is concave down and the point is a local maximum, whereas if it is positive, the function is concave up and the point is a local minimum.

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