Answer :
To find the intervals where the function is increasing or decreasing, we need to find the critical points. After finding the critical points and inflection points, we can determine the intervals of increasing, decreasing, concave up, concave down, and the local minimum and maximum.
To find the intervals where the function is increasing or decreasing, we need to find the critical points first. The critical points occur when the derivative of the function is equal to zero or does not exist. In this case, the derivative of f(x) is [tex]f'(x) = 10x^4 - 140x^3 + 450x^2.[/tex] Setting f'(x) = 0, we can solve for x to find the critical points. The local minimum or maximum occurs at these critical points.
To determine the intervals of concavity, we need to find the second derivative of f(x). The second derivative of f(x) is [tex]f''(x) = 40x^3 - 420x^2 + 900x.[/tex] Setting f''(x) = 0, we can solve for x to find the inflection points.
After finding the critical points and inflection points, we can determine the intervals where the function is increasing, decreasing, concave up, concave down, and the local minimum and maximum.
-∞ to the first critical point
From the first to the second critical point
Occurs at the first critical point
Occurs at the second critical point
-∞ to the first inflection point
From the first to the second inflection point
The points where the second derivative is equal to zero
To know more about critical points refer here:
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