Problem 30 - Calculus A New Horizon

Question

Give a complete graph of the function, and identify the location of all critical points and inflection poir s. Check your work with a graphing utility. $$4 x^{1 / 3}-x^{4 / 3}$$

Step-by-Step Solution

Answer

Critical points at \( x = 0 \) and \( x = 1 \); inflection point at \( x = 2 \).

Step 1: Find the first derivative

To find critical points, we need the first derivative of the function. The function is given by \[ f(x) = 4x^{1/3} - x^{4/3}. \]The first derivative, using power rules, is\[ f'(x) = \frac{4}{3}x^{-2/3} - \frac{4}{3}x^{1/3}. \]

Step 2: Solve for critical points

Critical points occur where the first derivative is zero or undefined. Set the first derivative to zero:\[ \frac{4}{3}x^{-2/3} - \frac{4}{3}x^{1/3} = 0. \]Factor out the common term:\[ \frac{4}{3}x^{-2/3}(1 - x) = 0. \]This gives two potential critical points: \( x = 0 \) and \( x = 1 \).

Step 3: Find the second derivative

To find inflection points, we need the second derivative. The first derivative is:\[ f'(x) = \frac{4}{3}x^{-2/3} - \frac{4}{3}x^{1/3}. \] Differentiating again, we find\[ f''(x) = -\frac{8}{9}x^{-5/3} + \frac{4}{9}x^{-2/3}. \]

Step 4: Solve for inflection points

Set the second derivative to zero to find potential inflection points:\[ -\frac{8}{9}x^{-5/3} + \frac{4}{9}x^{-2/3} = 0. \]Factor out the common term:\[ -\frac{4}{9}x^{-5/3}(2 - x) = 0. \]This implies that the second derivative changes sign at \( x = 2 \). Thus, the inflection point is \( x = 2 \).

Step 5: Verify with a graphing utility

Using a graphing utility, plot the function \( f(x) = 4x^{1/3} - x^{4/3} \). Check if the graph shows critical points at \( x = 0 \) and \( x = 1 \), and an inflection point at \( x = 2 \). The graph should display these points accurately as confirmed by the graphing utility.

Key Concepts

Critical Points

Critical points of a function are vital in determining where the function's graph has peaks and valleys. To find these, we focus on where the first derivative equals zero or is undefined. Here's a basic process to identify critical points:

• Find the first derivative of the function.
• Set this derivative equal to zero and solve for the variable.
• Examine where the derivative is undefined, as these can also be critical points.

For the function given \[ f(x) = 4x^{1/3} - x^{4/3}, \]we found that the first derivative is\[ f'(x) = \frac{4}{3}x^{-2/3} - \frac{4}{3}x^{1/3}. \]By setting \( f'(x) = 0 \)and solving, we discovered that \( x = 0 \) and \( x = 1 \)are the critical points. These points are significant because they indicate where the function might change from increasing to decreasing or vice versa.

Inflection Points

Inflection points are where a curve changes concavity, from concave up to concave down, or the other way around. To determine where these points are, we use the second derivative.

• Find the second derivative of the function.
• Set this second derivative equal to zero and solve.
• Check where the second derivative changes sign, confirming a change in concavity.

In our function, the second derivative was calculated as\[ f''(x) = -\frac{8}{9}x^{-5/3} + \frac{4}{9}x^{-2/3}. \]By setting \( f''(x) = 0 \),we determined that\( x = 2 \)is an inflection point. This point is essential for understanding the function's curvature, showing a transition in the graph's concave structure.

Graphing Utilities

Graphing utilities are powerful tools for checking the accuracy of analytical solutions. They allow us to visualize a function and inspect critical and inflection points on its graph effortlessly. Here’s how graphing utilities can aid our process:

• Plot the function using a graphing tool.
• Visually identify critical points where the slope of the tangent is zero (peaks and troughs).
• Spot inflection points where the graph changes concavity.

For our function \( f(x) = 4x^{1/3} - x^{4/3}, \)a graph will show critical points at \( x = 0 \) and \( x = 1 \),as well as an inflection point at \( x = 2 \).Using a graphing utility confirms these points, providing a visual method of verification. This supportive approach helps cement your understanding of differential calculus concepts and their graphical representation.