Questions: Find the inflection point(s) for the function (f(x)=5 x^2/3-x^5/3-3) a.) ((-1,-5)) b.) ((-1,3)) and ((0,-3)) c.) ((0,-3)) d.) ((-1,3))

Transcript text: Find the inflection point(s) for the function $f(x)=5 x^{\frac{2}{3}}-x^{\frac{5}{3}}-3$ a.) $(-1,-5)$ b.) $(-1,3)$ and $(0,-3)$ c.) $(0,-3)$ d.) $(-1,3)$

Solution

Solution Steps

To find the inflection points of the function $ f(x) = 5x^{\frac{2}{3}} - x^{\frac{5}{3}} - 3 $, we need to determine where the second derivative changes sign. This involves the following steps:

Compute the first derivative $ f'(x) $.
Compute the second derivative $ f''(x) $.
Solve $ f''(x) = 0 $ to find potential inflection points.
Check the sign change of $ f''(x) $ around these points to confirm they are inflection points.

Step 1: Compute the First Derivative

The first derivative of the function $ f(x) = 5x^{\frac{2}{3}} - x^{\frac{5}{3}} - 3 $ is calculated as follows: \[ f'(x) = \frac{10}{3} x^{-\frac{1}{3}} - \frac{5}{3} x^{\frac{2}{3}} \]

Step 2: Compute the Second Derivative

Next, we compute the second derivative: \[ f''(x) = -\frac{10}{9} x^{-\frac{4}{3}} - \frac{10}{9} x^{-\frac{1}{3}} \]

Step 3: Find Potential Inflection Points

To find the potential inflection points, we set the second derivative equal to zero: \[ -\frac{10}{9} x^{-\frac{4}{3}} - \frac{10}{9} x^{-\frac{1}{3}} = 0 \] This simplifies to: \[ x^{-\frac{4}{3}} + x^{-\frac{1}{3}} = 0 \] The solution yields $ x = -1 $.

Step 4: Evaluate the Function at the Inflection Point

We evaluate the function at $ x = -1 $: \[ f(-1) = 5(-1)^{\frac{2}{3}} - (-1)^{\frac{5}{3}} - 3 = 5(1) - (-1) - 3 = 5 + 1 - 3 = 3 \]