How do you sketch the graph #y=x^4x^3x# using the first and second derivatives?
See below
Using the power rule
This is a polynomial of degree 3. To find zeros for polynomials of degree 3 or higher we use Rational Root Test.
To find any other real roots:
To find other zeros:
graph{x^4x^3x [10, 10, 5, 5]}
By signing up, you agree to our Terms of Service and Privacy Policy
To sketch the graph of (y = x^4  x^3  x) using the first and second derivatives, follow these steps:
 Find the critical points by setting the first derivative equal to zero and solving for (x).
 Use the second derivative test to determine the nature of each critical point (whether it's a local minimum, maximum, or inflection point).
 Plot the critical points on the graph.
 Determine the behavior of the function as (x) approaches positive and negative infinity.
 Sketch the graph, incorporating the information from steps 14.
Let's go through each step:

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).

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.

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.

Determine the behavior of the function as (x) approaches positive and negative infinity by observing the leading term of the function (x^4).

Sketch the graph based on the information obtained from steps 14, 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).
By signing up, you agree to our Terms of Service and Privacy Policy
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.
 What is the inverse function of #f(x)=1/(1+x)#. How inverse functions appear in graph?
 Let #y = f(x)# be a twicedifferentiable function such that #f(1) = 2# and . What is the value of #(d^2y)/(dx^2)# at # x = 1#?
 How do you find the inflection points of the graph of the function: #f(x) = (x+2) (x4)^2#?
 What is the second derivative of #y=x*sqrt(16x^2)#?
 How do you find points of inflection for #y= sin x + cos x#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7