Local Maximum and Local Minimum of a Function

In order to understand the difference between local maximum and local minimum values of a function, let's first define what they mean.

Local Maximum: A local maximum refers to the highest point on a graph within a specific interval. It is the y-coordinate of a point where the function reaches its highest value in a small neighborhood around that point.

Local Minimum: A local minimum, on the other hand, is the lowest point on a graph within a specific interval. It is the y-coordinate of a point where the function reaches its lowest value in a small neighborhood around that point.

Function f(x) = 2x^3 - 24x + 107

Now, let's analyze the given function f(x) = 2x^3 - 24x + 107.

To find the local maximum and local minimum values of this function, we need to identify the critical points, which are the points where the derivative of the function is either zero or undefined.

Step 1: Find the derivative of the function
f'(x) = 6x^2 - 24

Step 2: Set the derivative equal to zero and solve for x
6x^2 - 24 = 0
x^2 - 4 = 0
(x - 2)(x + 2) = 0

Solving for x, we find two critical points: x = -2 and x = 2.

Step 3: Determine the nature of the critical points
To determine whether these critical points are local maximum or local minimum values, we can use the second derivative test.

Step 3.1: Find the second derivative of the function
f''(x) = 12x

Step 3.2: Evaluate the second derivative at the critical points
f''(-2) = 12(-2) = -24
f''(2) = 12(2) = 24

Step 3.3: Interpret the results
Since the second derivative at x = -2 is negative (-24), it indicates a concave downward shape, which means the function has a local maximum at x = -2.

Similarly, since the second derivative at x = 2 is positive (24), it indicates a concave upward shape, which means the function has a local minimum at x = 2.

Conclusion

In summary, the local maximum value of the function f(x) = 2x^3 - 24x + 107 occurs at x = -2, and the local minimum value occurs at x = 2. Therefore, the correct answer is option 'A' (64).