How do you sketch the graph #y=x^4-x^3-x# using the first and second derivatives?

Answer 1

See below

#y = x^4-x^3-x#

Using the power rule

#y' = 4x^3-3x^2-1#
#y'' = 12x^2-6x#
For turning points of #y#: #y'=0#
#:. 4x^3-3x^2-1 =0#

This is a polynomial of degree 3. To find zeros for polynomials of degree 3 or higher we use Rational Root Test.

The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction #p/q#, where p is a factor of the trailing constant and q is a factor of the leading coefficient.
In this case the factors of the leading coefficient #(4)# are #1,2,4 [q]# and factor of the trailing constant #(-1)# is #1 [p]#
Hence, the possible rational roots of #y'# are #+-1/1, +-1/2, +-1/4#
Testing each in turn reveals #y'(1) = 4-3-1=0#
Hence #x=1# is a rational root of #y' -> (x-1)# is a factor.

To find any other real roots:

#(4x^3-3x^2-1 )/(x-1) = 4x^2+x+1#
This is a quadtatic of the form: #ax^2+bx+c#
Test for real roots: #b^2-4ac>=0#
#1^2-4xx4xx1<0 -># the other two roots #in CC#
Hence, #x=1# is the only real root of #y'#
To test the nature of #y(1)#:
#y''(1) = 12-6 > 0 -> y(1) = y_min =-1#
#:. y# has a single minimum value of #-1# at #x=1#
By inspection it is clear that #y# has a zero at #x=0#

To find other zeros:

#x^3-x^2-1=0#
Unfortunately, this cubic has no rational roots and one real root at #x approx 1.46557# This result was obtained using an online polynomial root calculator: https://tutor.hix.ai
We now have the critical points of #y# shown on the graph below

graph{x^4-x^3-x [-10, 10, -5, 5]}

[NB: In practice, it would probably be necessary to plot a few extra points in the interval, say, #(-1.2,+2)# to produce this graph.]
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Answer 2

To sketch the graph of (y = x^4 - x^3 - x) using the first and second derivatives, follow these steps:

  1. Find the critical points by setting the first derivative equal to zero and solving for (x).
  2. Use the second derivative test to determine the nature of each critical point (whether it's a local minimum, maximum, or inflection point).
  3. Plot the critical points on the graph.
  4. Determine the behavior of the function as (x) approaches positive and negative infinity.
  5. Sketch the graph, incorporating the information from steps 1-4.

Let's go through each step:

  1. Find the first derivative of (y = x^4 - x^3 - x) and set it equal to zero: [ \frac{dy}{dx} = 4x^3 - 3x^2 - 1 = 0 ] Solve for (x).

  2. Once you find the critical points, (x_1), (x_2), and (x_3), evaluate the second derivative at these points: [ \frac{d^2y}{dx^2} = 12x^2 - 6x ] Evaluate ( \frac{d^2y}{dx^2} ) at each critical point.

  3. Use the second derivative test:

    • If ( \frac{d^2y}{dx^2} > 0 ), the critical point is a local minimum.
    • If ( \frac{d^2y}{dx^2} < 0 ), the critical point is a local maximum.
    • If ( \frac{d^2y}{dx^2} = 0 ), the test is inconclusive.
  4. Determine the behavior of the function as (x) approaches positive and negative infinity by observing the leading term of the function (x^4).

  5. Sketch the graph based on the information obtained from steps 1-4, including the critical points and the behavior of the function at positive and negative infinity. Make sure to accurately represent the concavity of the function based on the signs of the second derivative.

Once you've followed these steps, you should have a rough sketch of the graph of (y = x^4 - x^3 - x).

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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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