How do you find the intervals of increasing and decreasing using the first derivative given #y=(x-1)^2(x+3)#?

Answer 1

The function #y=(x-1)^2(x+3)# is rising in the interval #(-oo,-5/3)# and then it starts declining in the interval #(-5/3,1)# and then again rising in the interval #(1, oo)#

As #y=(x-1)^2(x+3)#
#(dy)/(dx)=2(x-1)xx(x+3)+(x-1)^2#
= #2(x^2+3x-x-3)+x^2-2x+1#
= #3x^2+2x-5#
= #3x^2-3x+5x-5#
= #3x(x-1)+5(x-1)#
= #(3x+5)(x-1)# whose zeros are #-5/3# and #1#

Currently utilizing a Sign Chart

#color(white)(XXXXXXXXXXX)-5/3color(white)(XXXXX)1#
#(x-1)color(white)(XXX)-ive color(white)(XXXX)-ive color(white)(XXXX)+ive#
#(3x+5)color(white)(XXX)-ive color(white)(XXXX)+ive color(white)(XXXX)+ive#
#(3x+5)(x-1)color(white)()+ive color(white)(XXX)-ive color(white)(XXXX)+ive#
Hence, the function #y=(x-1)^2(x+3)# is rising in the interval #(-oo,-5/3)# and then it starts declining in the interval #(-5/3,1)# and then again rising in the interval #(1, oo)#
#-5/3# amd #1# are local maxima and minima.

chart{(x-1)^2(x+3) [-5, 5, -10, 10]}

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Answer 2

To find the intervals of increasing and decreasing for the function (y = (x - 1)^2(x + 3)), we first find the first derivative, (y'), then determine where it is positive (increasing) or negative (decreasing).

Given (y = (x - 1)^2(x + 3)), let's first find (y'):

[y = (x - 1)^2(x + 3)] [y' = 2(x - 1)(x + 3) + (x - 1)^2]

Now, to find where (y') is positive or negative, we find the critical points by setting (y') equal to zero and solving for (x):

[0 = 2(x - 1)(x + 3) + (x - 1)^2]

Solve for (x) to find critical points. Then, use the first derivative test to determine the intervals of increasing and decreasing.

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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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