How do you find points of inflection and determine the intervals of concavity given #y=x^3-3x^2+4#?

Answer 1

concave down on #(-infinity,1) and concave up on (1, infinity)
x=1 is an inflection point.

  1. # y'=3x^2-6x#
  2. #y'' = 6x-6 or 6(x-1)#
  3. prepare sign chart with x=1 as a critical point using second derivative.
  4. to the left of x=1, y'' is negative so concave down. to right of x=1, it is positive so concave up.
  5. since the concavity changes at x=1, it is an inflection point.
Sign up to view the whole answer

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

Sign up with email
Answer 2

To find points of inflection and determine intervals of concavity for the function ( y = x^3 - 3x^2 + 4 ), follow these steps:

  1. Find the second derivative of the function.
  2. Set the second derivative equal to zero and solve for ( x ). These values are potential points of inflection.
  3. Test the intervals between these potential points of inflection using the second derivative test to determine concavity.

Let's go through these steps:

  1. Find the first derivative of the function: ( y' = 3x^2 - 6x ).
  2. Find the second derivative: ( y'' = 6x - 6 ).
  3. Set the second derivative equal to zero and solve for ( x ): ( 6x - 6 = 0 ) ( 6x = 6 ) ( x = 1 ).

So, ( x = 1 ) is a potential point of inflection. 4. Test the intervals ( (-\infty, 1) ) and ( (1, +\infty) ) using the second derivative test:

  • For ( x = 0 ), ( y'' = -6 < 0 ), indicating concave down.
  • For ( x = 2 ), ( y'' = 6 > 0 ), indicating concave up.

Therefore, ( x = 1 ) is a point of inflection, and the interval ( (-\infty, 1) ) is concave down, while the interval ( (1, +\infty) ) is concave up.

Sign up to view the whole answer

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

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7