How do you sketch the curve #y=x^3-3x^2-9x+5# by finding local maximum, minimum, inflection points, asymptotes, and intercepts?

Answer 1

graph{x^3-3x^2-9x+5 [-145.9, 172.6, -85.6, 73.6]} FIrst determine the interval of definition, then the behavior of first and second derivatives and the behavior of the function as #x#approaches #+-oo#

1) The function is a polynomial and is defined for #x in (-oo,+oo)#
  1. The highest monomial is of odd order so:
#lim_(x->-oo) y(x) = -oo# #lim_(x->+oo) y(x) = +oo#
Also #(y(x))/x# does not have a finite limit, so the function has no asymptotes.
  1. Calculate the first and second derivative:
#y'(x) = 3x^2-6x-9# #y''(x) = 6(x-1)#
The points where #y'(x) = 0# are:
#x=(3+-sqrt(9+27))/3 = (3+-6)/3#
#x_1=-1, x_2=3#

In both points the second derivative is non null, so these are local extrema and not inflection points.

If we analyze the sign of #y'(x)# we see that:
#y'(x) > 0# for #x in (-oo, -1) and x in (3, +oo)# #y'(x) < 0# for #x in (-1,3)#
So, #y(x)# starts from #-oo#, grows util #x=-1# where it reaches a local maximum, decreases until #x=3# where it reaches a local minimum, that starts growing again.
  1. Analyze the value of the function in the local extrema:
#y(x)_(x=-1) =(-1)^3-3(-1)^2-9(-1)+5 =-1-3+9+5=10#
#y(x)_(x=3) =3^3-3*3^2-9*3+5 =27-27-27+5=-22#
So there will be three real roots: one in #(-oo, -1)#, one in #(-1,3)# and one in #(3, +oo)#
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Answer 2

To sketch the curve ( y = x^3 - 3x^2 - 9x + 5 ) and identify its local maximum, minimum, inflection points, asymptotes, and intercepts, follow these steps:

  1. Intercepts:

    • y-intercept: Set ( x = 0 ) and solve for ( y ). ( (0, 5) ).
    • x-intercepts: Set ( y = 0 ) and solve for ( x ).
  2. Critical Points:

    • Find the first derivative ( y' ).
    • Set ( y' = 0 ) and solve for ( x ).
    • These are potential critical points.
  3. Second Derivative Test:

    • Find the second derivative ( y'' ).
    • Substitute critical points into ( y'' ) to determine concavity.
    • Use the Second Derivative Test to classify critical points as local maximum, minimum, or points of inflection.
  4. Asymptotes:

    • Vertical Asymptotes: Check for vertical asymptotes by finding where the denominator of rational functions is zero.
    • Horizontal Asymptotes: Determine the behavior of the function as ( x ) approaches positive and negative infinity.
  5. Sketching:

    • Plot the intercepts.
    • Use the behavior of the function near critical points and asymptotes to sketch the curve.
    • Label points as local maximum, minimum, or points of inflection based on the Second Derivative Test.

These steps will help you sketch the curve and identify its key features accurately.

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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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