Question

How can you verify, for a given value of the independent variable, where a maximum or minimum occurs using rate of change calculations?

a) By analyzing the sign of the first derivative: A maximum occurs where the first derivative changes from positive to negative, and a minimum occurs where it changes from negative to positive.
b) By calculating the second derivative: A maximum occurs when the second derivative is negative, and a minimum occurs when it is positive.
c) By finding the critical points and using the second derivative test: A maximum occurs at a critical point where the second derivative is negative, and a minimum occurs where it is positive.
d) By using the average rate of change at the given value and comparing it to nearby points.

Answer 1

To verify a local maximum or minimum using rate of change calculations, we use the sign of the first derivative or the value of the second derivative at critical points. The first derivative test checks for a sign change, while the second derivative test assesses concavity.

Step-by-step explanation:

The verification of a maximum or minimum value in a function, for a given value of the independent variable using rate of change calculations, can be done using several methods. The two most common methods include first derivative test and second derivative test.

1. First Derivative Test: This involves analyzing the sign of the first derivative of the function. If the first derivative changes from positive to negative at a point, it indicates a local maximum. If it changes from negative to positive, it indicates a local minimum.
2. Second Derivative Test: This involves calculating the second derivative of the function. A local maximum occurs where the second derivative is negative, and a local minimum occurs where the second derivative is positive. This is because a negative second derivative suggests the function is concave down (forming a peak), and a positive second derivative indicates the function is concave up (forming a valley).

It is also essential to identify the critical points, which are points where the first derivative is zero or undefined. The second derivative test is applied at these critical points to determine if they represent a maximum or minimum.