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Answer :
Final Answer:
To find the transition points, critical points, points of inflection, and local extrema of the function [tex]\(y = 3x^5 - 45x^3\)[/tex], we'll take its first and second derivatives and solve for where they equal zero. The critical points are [tex]\(x = -\sqrt{3}, \, 0, \, \sqrt{3}\)[/tex]. The points of inflection occur at [tex]\(x = -\sqrt{5}, \, 0, \,[/tex] \sqrt{5}\). There is a local maximum at[tex]\(x = -\sqrt{3}\)[/tex] and a local minimum at [tex]\(x = \sqrt{3}\)[/tex].
Explanation:
To find the transition points, critical points, points of inflection, and local extrema of the function [tex]\(y = 3x^5 - 45x^3\)[/tex], we employ differential calculus.
Firstly, we find the first derivative of the function to locate critical points where the slope of the function changes. Taking the derivative [tex]\(y'\), we get \(y' = 15x^4 - 135x^2\)[/tex]. Setting this derivative equal to zero and solving for x, we find the critical points. Solving[tex]\(15x^4 - 135x^2 = 0\), we get \(x = -\sqrt{3}, \, 0, \, \sqrt{3}\).[/tex]
Next, we find the second derivative[tex]\(y''\)[/tex]to determine points of inflection and ascertain the concavity of the function. The second derivative is[tex]\(y'' = 60x^3 - 270x\).[/tex] [tex]Setting \(y''\) equal to zero and solving for \(x\), we identify points of inflection. Solving \(60x^3 - 270x = 0\), we get \(x = -\sqrt{5}, \, 0, \, \sqrt{5}\).[/tex]
Finally, we determine the nature of the critical points by analyzing the concavity of the function. Around each critical point, if the concavity changes, it indicates a local extremum. By evaluating the sign of the second derivative at the critical points, we find that[tex]\(x = -\sqrt{3}\)[/tex]corresponds to a local maximum and[tex]\(x = \sqrt{3}\)[/tex] corresponds to a local minimum.
In summary, the transition points occur where the function changes direction, critical points represent points where the derivative is zero, points of inflection occur where the concavity changes, and local extrema denote local maxima or minima of the function.
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