Finding Points of Inflection In Exercises 13-34 find the points of inflection and discuss the concavity of the graph of the function. 33. f(x)=arcsinx^4/3

To find the points of inflection and discuss the concavity of the graph of the function \( f(x) = \arcsin(x^{4/3}) \), we need to follow these steps:1. Find the first derivative \( f'(x) \).2. Find the second derivative \( f''(x) \).3. Set the second derivative equal to zero and solve for \( x \) to find potential points of inflection.4. Determine the intervals where the function is concave up or concave down based on the sign of \( f''(x) \).5. Identify the points of inflection where the concavity changes.Let's go through each step in detail:

Step 1:

Find the first derivative \( f'(x) \)The derivative of \( \arcsin(u) \) with respect to \( u \) is \( \frac{1}{\sqrt{1-u^2}} \). Using the chain rule, we get:\[ f'(x) = \frac{d}{dx} \arcsin(x^{4/3}) = \frac{1}{\sqrt{1-(x^{4/3})^2}} \cdot \frac{d}{dx}(x^{4/3}) \]\[ f'(x) = \frac{1}{\sqrt{1-x^{8/3}}} \cdot \frac{4}{3}x^{1/3} \]\[ f'(x) = \frac{4}{3} \cdot \frac{x^{1/3}}{\sqrt{1-x^{8/3}}} \]

Step 2:

Find the second derivative \( f''(x) \)We need to differentiate \( f'(x) \) with respect to \( x \).\[ f''(x) = \frac{d}{dx} \left( \frac{4}{3} \cdot \frac{x^{1/3}}{\sqrt{1-x^{8/3}}} \right) \]Using the quotient rule, where \( u = x^{1/3} \) and \( v = \sqrt{1-x^{8/3}} \), we have:\[ f''(x) = \frac{4}{3} \left( \frac{d}{dx} u \cdot v - u \cdot \frac{d}{dx} v \right) \div v^2 \]\[ f''(x) = \frac{4}{3} \left( \frac{1}{3}x^{-2/3} \cdot \sqrt{1-x^{8/3}} - x^{1/3} \cdot \frac{-8}{3}x^{5/3} \right) \div \sqrt{1-x^{8/3}}^2 \]\[ f''(x) = \frac{4}{3} \left( \frac{1}{3}x^{-2/3}sqrt{1-x^{8/3}} - x^{1/3} \cdot \frac{-8}{3}x^{5/3} \right) \div (1-x^{8/3}) \]

Step 3:

Set the second derivative equal to zero and solve for \( x \)To find the points of inflection, we set \( f''(x) = 0 \) and solve for \( x \).\[ \frac{4}{3} \left( \frac{1}{3}x^{-2/3} \cdot \sqrt{1-x^{8/3}} - x^{1/3} \cdot \frac{-8}{3}x^{5/3} \right) \div (1-x^{8/3}) = 0 \]This equation is complex to solve algebraically, so we may need to use numerical methods or graphing techniques to find the values of \( x \) that satisfy this equation.

Step 4:

Determine the intervals where the function is concave up or concave downBased on the sign of \( f''(x) \), we can determine where the function is concave up (positive) or concave down (negative).

Step 5:

Identify the points of inflectionThe points of inflection occur where the concavity changes, which corresponds to the values of \( x \) found in Step 3.Accurate Answer:The points of inflection and the concavity of the graph of the function \( f(x) = \arcsin(x^{4/3}) \) can be determined by solving the second derivative equation numerically or graphically to find the exact values of \( x \) where the concavity changes. These points are where the function changes from concave up to concave down or vice versa.