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The first derivative of a function is crucial for finding critical points. These points are where the slope of the function, or the rate of change, is zero. To find the first derivative of the given function \( f(x) = \frac{x^{4}}{4} - \frac{5x^{3}}{3} - 4x^{2} + 48x \), we apply the power rule. This rule states that the derivative of \( x^n \) is \( nx^{n-1} \). Using this, we calculate:

• Derivative of \( \frac{x^4}{4} \) is \( x^3 \).
• Derivative of \( \frac{-5x^3}{3} \) is \( -5x^2 \).
• Derivative of \( -4x^2 \) is \( -8x \).
• Derivative of \( 48x \) is \( 48 \).

Thus, the first derivative is \( f'(x) = x^3 - 5x^2 - 8x + 48 \).
Setting this equal to zero gives us the critical points, where the function switch from increasing to decreasing or vice versa. These points are pivotal in determining key characteristics of the function like local maxima or minima.

The Second Derivative Test helps determine whether a critical point is a local maximum, local minimum, or neither. Once the critical points are identified, the test uses the second derivative of the function. This involves evaluating whether the second derivative is positive, negative, or zero at these points.
For the function \( f(x) \), the second derivative is found by taking the derivative of the first derivative: \( f''(x) = 3x^2 - 10x - 8 \).

• If \( f''(x) > 0 \) at a critical point, the function is concave up, indicating a local minimum.
• If \( f''(x) < 0 \), it is concave down, indicating a local maximum.
• If \( f''(x) = 0 \), the test is inconclusive, and further analysis is needed.

Using this test simplifies determining the nature of the critical points compared to other methods of graph analysis.

A local maximum refers to a point on the graph where a function changes from increasing to decreasing. This point is higher than any nearby points. To find a local maximum, utilize the second derivative test.
In our function, we evaluate the second derivative at the critical point \( x \approx 1.608 \). Here, \( f''(1.608) \approx -3.144 \), which is less than zero. This negative value signals a local maximum because the graph is concave down at this point.

• The critical point \( x \approx 1.608 \) is thus confirmed as a local maximum.
• The negative second derivative value at this point corroborates that the function decreases after reaching a peak here.

Identifying local maxima is helpful in understanding where a function's highest turning point occurs within a given region.

A local minimum occurs where a function changes from decreasing to increasing, creating a low point compared to surrounding values. The Second Derivative Test aids in identifying these points by checking if the second derivative is positive at the critical point.
In our case, calculate \( f''(5.392) \approx 7.144 \), which is greater than zero. This outcome implies that the function is concave up around \( x \approx 5.392 \), suggesting this is a local minimum.

• The point \( x \approx 5.392 \) stands at a local trough in the function's graph.
• The positive second derivative ensures that the function shifts from decreasing to increasing at this point.

This understanding of local minima helps in mapping out the function's lower turn, thus aiding in the broader picture of the function's behavior.