Answer :
Final answer:
To find transition points like local minimum and points of inflection for the function y = 5x^6 - 45x^4, one must calculate the first and second derivatives, find where the first derivative equals zero, and then use the second derivative to test for concavity.
Explanation:
The student is asking about finding the transition points, such as local minimum and points of inflection, of the polynomial function y = 5x6 - 45x4. To locate these points, one would first find the first derivative of the function to determine where it equals zero, which would indicate potential local extrema. The second step is to find the second derivative to determine concavity and identify points of inflection.
To find a local minimum, we would set the first derivative equal to zero and solve for x. Any critical points found would then be tested with the second derivative to determine if they are maxima, minima, or points of inflection. A positive second derivative indicates a local minimum, while a negative second derivative indicates a local maximum. If the second derivative equals zero, further investigation is required to determine if it's a point of inflection.
