How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for # 3x^5 - 5x^3#?

Answer 1

Take the first derivative...

#f'(x) = 15x^4 - 15x^2#
Minima and maxima occur at places where the above equation evaluates to zero. Right away, you should be able to see that #x = 0# is one of these places.

But where else?

#f'(x) = 15x^2(x^2 - 1)#
#=15x^2(x+ 1)(x - 1)#
...which gives you the other points: #+1 and -1#

To determine whether these might be maxima or minima, you must take the second derivative:

#f''(x) = 60x^3 - 30x#

and evaluate at x = -1, 0, and +1.

#f''(-1) = -60 + 30 = -30#, which is negative, so #x = -1# is a relative maxima.
#f''(1) = 30#, which is positive, so #x = 1# is a relative minima.
#f''(0) = 0#, so this point is neither minima nor maxima.

Finding the regions where the original function is increasing or decreasing requires a little more analysis:

Examine the first derivative equation:

( Eq. 1) #f'(x) = 15x^2(x+ 1)(x - 1)#
Note that if x < -1, then the terms x+1 and x-1 are both negative, and term #15x^2# is positive, since any number squared is positive.
Therefore, in the region #x < -1#, the first derivative evaluates to a positive * negative * negative, and is therefore positive. So the original function is increasing where #x < -1#.
In the region x > 1, the second derivative is obviously positive, so the function is increasing where #x > 1#.
In the region #-1 < x < 0#, the #x + 1# term is positive, and the #x-1# term is negative. The first derivative is therefore a positive * negative * positive number, which is negative. The original function is therefore DECREASING in the region #-1 < x < 0#.
(a similar line of reasoning applies to the region #0 < x < 1#).

Always helps to have a graph of the function to serve as a "sanity check". graph{3x^5 - 5x^3 [-10, 10, -5, 5]}

GOOD LUCK!

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Answer 2

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:

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

  2. Set (f'(x) = 0) to find critical points. (15x^4 - 15x^2 = 0)
    (15x^2(x^2 - 1) = 0)
    (x = 0, \pm 1)

  3. 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.
  4. 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.
  5. 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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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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