91Ó°ÊÓ

Critical points are specific values of \( x \) where the derivative of the function \( f(x) \) is zero or undefined. These points are crucial as they can tell us a lot about the function's behavior, especially concerning local maximums and minimums. In the case of the function \( f(x) = \frac{1}{2} x^{2/3} - x^{1/3} \), the critical point arises when \( f'(x) = 0 \). Calculating the derivative gives:

• \( f'(x) = \frac{1}{3}x^{-1/3} - \frac{1}{3}x^{-2/3} \)

By setting this to zero, we find it simplifies to:

• \( \frac{1}{3}x^{-1/3} - \frac{1}{3}x^{-2/3} = 0 \)

This leads us to the critical point at \( x = 1 \). This point is noteworthy because it's often where the graph changes direction from increasing to decreasing, or vice versa. Hence, critical points help identify potential peaks and troughs in the graph.

Concavity helps us understand the curvature of a function. It describes whether the graph bends upwards (concave up) or downwards (concave down). We check concavity by examining the second derivative \( f''(x) \). For our function, the second derivative is:

• \( f''(x) = -\frac{1}{9}x^{-4/3} + \frac{2}{9}x^{-5/3} \)

Inflection points occur where the concavity changes. Solving \( f''(x) = 0 \):

• This gives \( x = 8 \), indicating a switch in concavity at this point.

By evaluating \( f''(x) \) on intervals around \( x = 8 \), we see:

• For \( x < 8 \), \( f''(x) > 0 \), implying the function is concave up.
• For \( x > 8 \), \( f''(x) < 0 \), implying the function is concave down.

Thus, \( x = 8 \) is an inflection point, marking a transition in the graph's bending behavior. This understanding helps us sketch more accurate and visually representative graphs.

Asymptotes are lines that the graph of a function approaches but never touches. They can be horizontal, vertical, or oblique. For the function \( f(x) = \frac{1}{2} x^{2/3} - x^{1/3} \), we particularly look for horizontal and vertical asymptotes. Horizontal asymptotes occur at limits as \( x \to \pm \infty \). Here:

• \( \lim_{x \to \infty} (\frac{1}{2}x^{2/3} - x^{1/3}) = 0 \)

This indicates a horizontal asymptote at \( y = 0 \). Vertical asymptotes occur where the function is undefined, often at points where the denominator is zero in a rational function. However, in this exercise, the function is defined for all \( x > 0 \) due to the real-valued powes, so there are no vertical asymptotes. Understanding these invisible boundaries helps predict the end behavior of a function's graph.

The behavior of a graph over specific intervals, either increasing or decreasing, can be determined using the first derivative of the function. Recall that the derivative \( f'(x) \) informs us of these intervals:

• It is negative in regions where the function is decreasing.
• It is positive in regions where the function is increasing.

In our function, by examining \( f'(x) \), we determined:

• The function is decreasing on \((0, 1)\).
• The function is increasing on \((1, \infty)\).

Interpreting these intervals accurately allows us to detect the overall trend of the function over its domain. These insights reflect changes in direction, marked by critical points, that are valuable in sketching and understanding the graph's visual trends.