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The First Derivative Test helps to determine whether a critical point is a relative maximum, minimum, or neither. Begin by taking the first derivative of the function. For example, given the function \( f(x) = 1 + 8x - 3x^2 \), the first derivative is \( f'(x) = 8 - 6x \). Critical points occur where \( f'(x) = 0 \). Solving \( 8 - 6x = 0 \), we find \( x = \frac{4}{3} \), which is a critical point.Next, observe the sign change of \( f'(x) \) around this critical point:

• If \( f'(x) \) changes from positive to negative, the critical point is a relative maximum.
• If it changes from negative to positive, the critical point is a relative minimum.

For \( f(x) = 1 + 8x - 3x^2 \), the first derivative changes from positive to negative around \( x = \frac{4}{3} \), indicating a local maximum.

The Second Derivative Test is a powerful tool to confirm the nature of a critical point found using the first derivative. It involves taking the second derivative of the original function.With \( f(x) = 1 + 8x - 3x^2 \), the second derivative is \( f''(x) = -6 \). Notice how it's a constant and negative, indicating the function is concave downward everywhere.The test involves evaluating the second derivative at the critical points found in the first derivative test:

• If \( f''(x) > 0 \), it suggests the function is concave up, signifying a local minimum.
• If \( f''(x) < 0 \), as in this example, it implies a concave down shape, confirming a local maximum.

Thus, for \( x = \frac{4}{3} \), since \( f''(\frac{4}{3}) = -6 \), we confirm a local maximum.

Critical points are the values of \( x \) where the first derivative is zero or undefined. These points are crucial in identifying the function's relative extrema, or local maximums and minimums.To find critical points from the function \( f(x) = 1 + 8x - 3x^2 \), first, calculate the derivative \( f'(x) = 8 - 6x \). Set \( f'(x) = 0 \) and solve for \( x \). The result is \( x = \frac{4}{3} \), the critical point of interest here.Knowing where critical points occur allows us to examine the behavior of the function around those points. By using first and second derivative tests, you can characterize these points to determine whether they correspond to local maxima, minima, or saddle points.