How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #y=x^42x^3#?
intercept / stationary point at x= 0 is an inflexion point
stationary point at
so the intercept and stationary point at x= 0 is an inflexion point
you can see all of this in the plot graph{x^4  2x^3 [10, 10, 5, 5]}
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To determine where the function ( y = x^4  2x^3 ) is increasing or decreasing, and to find relative maxima and minima:

Find the first derivative of the function, ( y' ), using the power rule. [ y' = 4x^3  6x^2 ]

Set ( y' = 0 ) to find critical points. [ 4x^3  6x^2 = 0 ] Factor out ( 2x^2 ): [ 2x^2(2x  3) = 0 ] So, ( x = 0 ) and ( x = \frac{3}{2} ) are critical points.

Determine the intervals between these critical points and test points within these intervals to determine the sign of ( y' ).
For ( x < 0 ): Choose ( x = 1 ) (test point). [ y'(1) = 4(1)^3  6(1)^2 = 4  6 = 10 ] Since ( y'(1) < 0 ), the function is decreasing on this interval.
For ( 0 < x < \frac{3}{2} ): Choose ( x = 1 ) (test point). [ y'(1) = 4(1)^3  6(1)^2 = 4  6 = 2 ] Since ( y'(1) < 0 ), the function is decreasing on this interval.
For ( x > \frac{3}{2} ): Choose ( x = 2 ) (test point). [ y'(2) = 4(2)^3  6(2)^2 = 32  24 = 8 ] Since ( y'(2) > 0 ), the function is increasing on this interval.

To find relative maxima and minima, examine the behavior of the function around the critical points.
At ( x = 0 ): Since the function changes from decreasing to increasing at this point, it is a relative minimum.
At ( x = \frac{3}{2} ): Since the function changes from increasing to decreasing at this point, it is a relative maximum.
Therefore, the function ( y = x^4  2x^3 ) has a relative minimum at ( x = 0 ) and a relative maximum at ( x = \frac{3}{2} ).
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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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