How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for # 3x^5  5x^3#?
Take the first derivative...
But where else?
To determine whether these might be maxima or minima, you must take the second derivative:
and evaluate at x = 1, 0, and +1.
Finding the regions where the original function is increasing or decreasing requires a little more analysis:
Examine the first derivative equation:
Always helps to have a graph of the function to serve as a "sanity check". graph{3x^5  5x^3 [10, 10, 5, 5]}
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To determine where the function (f(x) = 3x^5  5x^3) is increasing or decreasing and to find relative maxima and minima, follow these steps:

Find the first derivative of the function (f(x)). (f'(x) = 15x^4  15x^2)

Set (f'(x) = 0) to find critical points. (15x^4  15x^2 = 0)
(15x^2(x^2  1) = 0)
(x = 0, \pm 1) 
Use the first derivative test or the sign test to determine the intervals where the function is increasing or decreasing:
 Test a value less than 1 (e.g., 2) into (f'(x)) to find the sign in the interval (( \infty, 1)). It yields a positive value.
 Test a value between 1 and 0 (e.g., 0.5) into (f'(x)) to find the sign in the interval ((1, 0)). It yields a negative value.
 Test a value between 0 and 1 (e.g., 0.5) into (f'(x)) to find the sign in the interval ((0, 1)). It yields a positive value.
 Test a value greater than 1 (e.g., 2) into (f'(x)) to find the sign in the interval ((1, \infty)). It yields a positive value.

Use the second derivative test or evaluate (f''(x)) to confirm maxima and minima. (f''(x) = 60x^3  30x)
 Evaluate (f''(1)): (f''(1) = 60(1)^3  30(1) = 30), indicating a maximum.
 Evaluate (f''(0)): (f''(0) = 60(0)^3  30(0) = 0), inconclusive.
 Evaluate (f''(1)): (f''(1) = 60(1)^3  30(1) = 30), indicating a minimum.

Therefore, (x = 1) is a relative maximum and (x = 1) is a relative minimum.
In summary:
 The function is increasing on (( \infty, 1)) and ((1, \infty)).
 The function is decreasing on ((1, 0)) and ((0, 1)).
 Relative maximum occurs at (x = 1) and relative minimum occurs at (x = 1).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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