Math  /  Calculus

QuestionConsider the function m(x)=24x5360x4+1400x35.m(x)=24 x^{5}-360 x^{4}+1400 x^{3}-5 .
Differentiate mm and use the derivative to determine each of the following.
The intervals on which mm is increasing. mm increases on: \square The intervals on which mm is decreasing. mm decreases on: \square The value(s) of xx at which mm has a relative maximum. If there are more than one solutions, separate them by a comma. Use exact values. mm has local maximum(s) at x=x= \square The value(s) of xx at which mm has a relative minimum. If there are more than one solutions, separate them by a comma. Use exact values. mm has local minimum(s) at x=x= \square

Studdy Solution
Now, let's classify the critical points as maxima or minima:
At x = 0: The sign of m'(x) changes from positive to negative, so this is a relative maximum. At x = 5: The sign of m'(x) changes from negative to positive, so this is a relative minimum. At x = 7: The sign of m'(x) changes from positive to negative, so this is a relative maximum.
Now, let's answer the specific questions:
1. The intervals on which m is increasing: (-∞, 0) ∪ (7, ∞)
2. The intervals on which m is decreasing: (0, 5) ∪ (5, 7)
3. The value(s) of x at which m has a relative maximum: x = 0, 7
4. The value(s) of x at which m has a relative minimum: x = 5

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