Max/Min & Inflection Point of f(x)=(x+1)^2(x-2)

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In summary, for the function f(x)=(x+1)^2(x-2), the max and min values occur at x=-1 and x=1 respectively. To find the inflection point of concavity, we need to find where the second derivative of the function is equal to 0. This occurs at x=0, but we also need to check that the first derivative changes sign at this point to confirm it as an inflection point.
  • #1
fazal
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Homework Statement



let f(x)=(x+1)^2(x-2)
a)Find the max and Min values of f(x)
b)Find the inflection point of concavity

Homework Equations



using defferentiation



The Attempt at a Solution


for part a) differentiate (x+1)^2(x-2) we get ans=3(x-1)(x+1) Plse check for me?
than f'(x)=0 therefore the points i got is x=1 and x=-1 respective??


the second part not sure..plse assist.
 
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  • #2
You are right about the local max and min. For the second part--all an inflection point is, is where the second derivative (that which we use to determine concavity) is 0. So you would have

[tex]f''(x)=2(x-2)+4(x+1)=0[/tex]

Can you get it from there?
 
  • #3
Make sure you answer the question asked! Max and Min occur at x= -1 and x= 1 but you haven't yet said what the max and min are.

Also what jeffreydk said about the inflection point is slightly misleading. An inflection point is a point where the first derivative changes sign. That means the the second derivative must be 0 there but that is not sufficient. You need to check that the first derivative really does change sign. For example if f(x)= x4, f'= 4x3 and f"= 12x2. f"(0)= 0 but f' does NOT change sign there so (0,0) is NOT an inflection point of f(x)= x4.
 

1. What is the maximum or minimum point of the function f(x)=(x+1)^2(x-2)?

The maximum or minimum point of a function is also known as the "extremum" point. In this specific function, there is only one extremum point, which is known as the minimum point. It is located at x=-1, where the function reaches its lowest value of f(-1)=-9.

2. How do you find the maximum or minimum point of a function?

To find the maximum or minimum point of a function, you must first take the derivative of the function and set it equal to 0. Then, solve for the x-values that make the derivative equal to 0. These x-values will be the coordinates of the extremum point.

3. What is an inflection point?

An inflection point is a point on the graph of a function where the concavity changes. It is where the function changes from being concave up to concave down, or vice versa. It can also be described as the point where the second derivative of the function changes sign.

4. How can you determine if a function has an inflection point?

To determine if a function has an inflection point, you must take the second derivative of the function. If the second derivative changes sign at a specific x-value, then that x-value is an inflection point. In the function f(x)=(x+1)^2(x-2), the second derivative is 12x-4. It changes sign at x=1/3, so that is the inflection point.

5. Can a function have more than one maximum or minimum point?

Yes, a function can have multiple maximum or minimum points. However, in the function f(x)=(x+1)^2(x-2), there is only one minimum point. This is because the function is a cubic function, which has a basic shape of a curve with one minimum point. Functions with different shapes, such as quadratic functions, can have multiple maximum or minimum points.

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