91Ó°ÊÓ

To start finding the maximum value of a function, calculating the derivative is essential. The derivative tells us how the function changes at any given point.
For our function, which is given by:

Critical points are where the function changes direction, which means they're spots where the derivative is zero or undefined.
Finding them is necessary to determine where a function might have a maximum or minimum.

For our function, we set the derivative to zero and solve for x:
This equation is straightforward to solve:

2x = 12
x = 6

To confirm if a critical point is a maximum or minimum, we use the second derivative test.
We compute the second derivative of the function and check its sign at the critical point.
For our function, the second derivative is:
f''(x) = -2
If the second derivative is positive at the critical point, it's a local minimum.
If it's negative, we've got a local maximum.
In our case,
f''(6) = -2, which is less than 0
, confirming we have a maximum.

A quadratic function is a type of polynomial function with a degree of two.
Its general form is: ax² + bx + c

Quadratic functions form parabolas when graphed. They either open upwards (if 'a' is positive) or downwards (if 'a' is negative).
Our function here is a downward opening parabola since the coefficient of x² is negative (-1).
This is why we look for a maximum value.

To find this maximum (or minimum), follow these steps:
1. Find the first derivative to locate the critical points.
2. Use the second derivative test to determine the nature of the critical points.
3. Evaluate the quadratic function at the critical points.



For our function:

1. Derivative is f'(x)=12-2x
2.Second derivative is Ñ„''(x)=(-2),
f(x) at x=6 gives maximum value 36.