# What does the 2nd Derivative Test tell you about the behavior of #f(x) = x^4(x-1)^3# at these critical numbers?

The Second Derivative Test implies that the critical number (point) *saying nothing* about the nature of

Using the Product Rule again gives:

By signing up, you agree to our Terms of Service and Privacy Policy

The Second Derivative Test tells us about the behavior of a function at critical points by analyzing the concavity of the function. For the function ( f(x) = x^4(x-1)^3 ), we first find the critical points by setting the first derivative equal to zero and solving for ( x ). Then, we evaluate the second derivative at these critical points.

The critical points for ( f(x) ) occur where ( f'(x) = 0 ). So, we find ( f'(x) ) by taking the derivative of ( f(x) ) and setting it equal to zero: [ f'(x) = 4x^3(x-1)^3 + x^4(3)(x-1)^2 = 0 ]

After solving for ( x ), we obtain the critical points. Then, we evaluate the second derivative ( f''(x) ) at each critical point.

The second derivative of ( f(x) ) is: [ f''(x) = 12x^2(x-1)^3 + 12x^2(x-1)^2 + 4x^3(3)(x-1)^2 + 2x^4(3)(x-1) ]

After evaluating ( f''(x) ) at each critical point, we can determine the behavior of ( f(x) ) at those points:

- If ( f''(x) > 0 ) at a critical point, then the function is concave up at that point, indicating a local minimum.
- If ( f''(x) < 0 ) at a critical point, then the function is concave down at that point, indicating a local maximum.
- If ( f''(x) = 0 ) at a critical point, the test is inconclusive.

This information helps us understand the behavior of ( f(x) ) around its critical points and whether those points correspond to local maxima or minima.

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- What are the points of inflection of #f(x)=x^2-x^(1/2) #?
- What are the points of inflection of #f(x)=3ln(x^(2) +2) -2x #?
- For what values of x is #f(x)=(4x)/(x^2-1)# concave or convex?
- How do you find the inflection points of the graph of the function: #f(x)=x^4-6x^3#?
- What are the points of inflection, if any, of #f(x)=3x^5 - 5x^4 #?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7