How do you find all critical point and determine the min, max and inflection given #f(x)=3x^2-4x+1#?

Answer 1

Critical point x=#2/3#
Minima #-1/3#
No maxima, no inflection point.

Find f'(x) = 6x-4. Equate it to 0 to get #x=2/3#. This is the only critical point.
Now find f"(x) =6, which is a positive number (irrespective of any x) . Hence, there is a minima at x=2/3. The minima would be #3(2/3)^2 -4(2/3) +1#
= #-1/3#

Maxima does not exist.

Since f"(x) #!=# 0, there is no inflection point.
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Answer 2

To find all critical points, we first find the derivative of the function, ( f'(x) ). Then, we set ( f'(x) = 0 ) and solve for ( x ). The critical points are the ( x ) values where the derivative is zero or undefined.

Next, to determine whether each critical point corresponds to a minimum, maximum, or inflection point, we examine the sign of the second derivative, ( f''(x) ), at each critical point. If ( f''(x) > 0 ), the function has a local minimum at that critical point. If ( f''(x) < 0 ), the function has a local maximum at that critical point. If ( f''(x) = 0 ), the test is inconclusive, and we need further investigation.

To find inflection points, we locate the points where the concavity changes, which occurs where ( f''(x) = 0 ) or is undefined.

Let's follow these steps:

  1. Find ( f'(x) ) by differentiating ( f(x) = 3x^2 - 4x + 1 ).
  2. Set ( f'(x) = 0 ) and solve for ( x ) to find critical points.
  3. Compute ( f''(x) ) by differentiating ( f'(x) ).
  4. Evaluate ( f''(x) ) at each critical point to determine the nature of the critical points.
  5. Identify any inflection points by finding where ( f''(x) = 0 ) or is undefined.

Following these steps will help us find the critical points, as well as determine whether they correspond to minima, maxima, or inflection points.

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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.

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