91Ó°ÊÓ

To find relative extrema, we start with the first derivative. The first derivative of a function tells us the slope of the tangent line at any point on the graph. For a function\( f(x) = 5 - 12x - 2x^2 \), we find the first derivative by differentiating each term:
- The derivative of 5 is 0, since it's a constant.
- The derivative of \( -12x \) is \( -12 \).
- The derivative of \( -2x^2 \) is \( -4x \).
So, the first derivative\( f'(x) = -12 - 4x \). Understanding the first derivative helps us identify where the slopes of the line change, which is where critical points may be located.

Critical points occur where the first derivative is zero or undefined. These points are important because they can indicate where a function changes from increasing to decreasing, or vice versa. For our function, we set the first derivative to zero to find critical points:
\( -12 - 4x = 0 \)
By solving this equation, we get:
\( x = -3 \).
This value of x is a critical point. Critical points are essential for further determining whether a point is a maximum, minimum, or a saddle point.

The second derivative test helps us classify the critical points as either maxima, minima, or neither. We first need to find the second derivative of the function:
For the given function, \( f(x) = 5 - 12x - 2x^2 \), the first derivative is \( f'(x) = -12 - 4x \). Taking the derivative of \( -12 - 4x \) with respect to x gives us the second derivative:
\( f''(x) = -4 \)
The second derivative tells us about the concavity of the function. If the second derivative is positive at a critical point, the function is concave up, indicating a relative minimum. If it's negative, the function is concave down, indicating a relative maximum. Here, \( f''(x) = -4 \) is negative, so we have a relative maximum at \( x = -3 \).

Concavity describes the direction that a function curves. If a function is concave up, it curves upwards like a smile, and if it's concave down, it curves downwards like a frown. Concavity can be determined using the second derivative of the function.
For the given function, the second derivative \( f''(x) = -4 \) is constant and negative, showing that the function is concave down everywhere. This indicates that the graph of the function looks like an upside-down parabola, causing the critical point\( x = -3 \) to be a peak or a relative maximum. By understanding concavity, we can better visualize and anticipate the shape and behaviour of the graph of the function.