How do you find the maximum, minimum and inflection points and concavity for the function #y" = 6x  12#?
This is a straight line.
It's a line with no concavity or inflection points.
By signing up, you agree to our Terms of Service and Privacy Policy
To find the maximum, minimum, and inflection points, as well as determine the concavity of the function y'' = 6x  12, we need to follow these steps:

Take the second derivative of the given function to find the concavity: y'' = 6.

Since the second derivative is a constant (6), it indicates that the function is concave up everywhere.

To find the maximum or minimum points, we set the first derivative equal to zero and solve for x: y' = 3x^2  12x = 0 Factoring out 3x, we get: 3x(x  4) = 0 This yields two critical points: x = 0 and x = 4.

To determine whether these critical points correspond to a maximum, minimum, or neither, we use the second derivative test:
 For x = 0: Plug into the second derivative (y'' = 6x  12): y''(0) = 6(0)  12 = 12. Since the result is negative, it indicates a maximum point at x = 0.
 For x = 4: Similarly, y''(4) = 6(4)  12 = 12  12 = 0. The second derivative test is inconclusive, so we need additional information.

To find inflection points, we set the second derivative equal to zero and solve for x: 6x  12 = 0 Solving for x, we get: x = 2.

To confirm whether x = 4 is an inflection point, we check the concavity on either side:
 For x < 2, y'' < 0 (concave down).
 For x > 2, y'' > 0 (concave up). Thus, x = 2 is an inflection point.
To summarize:
 The function has a maximum point at (0, 12).
 It has an inflection point at (2, f(2)).
 Since the function is concave up everywhere, there are no minimum points.
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 xcoordinate of the point of inflection on the graph of #y = (1/3)x^3 + 5x^2+ 24#?
 Is #f(x)=sinx# concave or convex at #x=(3pi)/2#?
 How do you graph #f(x)=1/x3x^3# using the information given by the first derivative?
 How do you use the first and second derivatives to sketch #f(x)= x^4  2x^2 +3#?
 What are the points of inflection, if any, of #f(x) =(x+4)/(x2)^2#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7