find point of inflexion of x=(log y)³
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find point of inflexion of x=(log y)³
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Answer:
To find the point of inflection of the function x = (log y)³, we need to determine the second derivative and find the values of y where the second derivative is equal to zero or undefined.
First, let's find the first derivative of x with respect to y:
\( \frac{dx}{dy} = \frac{d}{dy}[(\log y)^3] \)
Using the chain rule, we have:
\( \frac{dx}{dy} = 3(\log y)^2 \cdot \frac{1}{y} \)
Now, let's find the second derivative by differentiating the first derivative with respect to y:
\( \frac{d^2x}{dy^2} = \frac{d}{dy}\left[3(\log y)^2 \cdot \frac{1}{y}\right] \)
Using the product rule and the chain rule, we have:
\( \frac{d^2x}{dy^2} = 3 \cdot 2(\log y) \cdot \frac{1}{y} \cdot \frac{1}{y} - 3(\log y)^2 \cdot \frac{1}{y^2} \)
Simplifying further:
\( \frac{d^2x}{dy^2} = \frac{6(\log y)}{y^2} - \frac{3(\log y)^2}{y^2} \)
To find the point of inflection, we need to set the second derivative equal to zero and solve for y:
\( \frac{6(\log y)}{y^2} - \frac{3(\log y)^2}{y^2} = 0 \)
Factoring out \( \frac{3(\log y)}{y^2} \), we get:
\( \frac{3(\log y)}{y^2} \left(2 - (\log y)\right) = 0 \)
This equation is satisfied when either \( \frac{3(\log y)}{y^2} = 0 \) or \( 2 - (\log y) = 0 \).
For \( \frac{3(\log y)}{y^2} = 0 \), we have \( \log y = 0 \), which implies \( y = 1 \).
For \( 2 - (\log y) = 0 \), we have \( \log y = 2 \), which implies \( y = 10^2 = 100 \).
Therefore, the points of inflection for the function x = (log y)³ are (1, x) and (100, x).