How do you find stationary and inflection points for #x^2-2x-3#?

Answer 1

The unique stationary point is #(1,-4)#. There are no inflection points.

When the derivative is zero at a point, it is referred to as "stationary"; these points are also known as critical points and are typically local extreme or "turning" points, though they are not always.

When #f(x)=x^2-2x-3# we get #f'(x)=2x-2# so that #x=1# is the first coordinate of the unique stationary point of this function (it also happens to be the lowest point on the graph). The second coordinate of this point is #f(1)=1-2-3=-4# so that you can think of the stationary point as being #(1,f(1))=(1,-4)#.
A point is an "inflection point" if the second derivate changes sign there. Since #f''(x)=2# for all #x#, the second derivative never changes sign. There are no inflection points for this function (the graph never changes from concave up (smile-like) to concave down (frown-like) or vice versa...in this case, the graph is always concave up).
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Answer 2

To find stationary points, you need to find the critical points by taking the derivative of the function and setting it equal to zero. Then solve for ( x ).

Given the function ( f(x) = x^2 - 2x - 3 ):

  1. Find the derivative ( f'(x) ). [ f'(x) = 2x - 2 ]

  2. Set ( f'(x) ) equal to zero and solve for ( x ). [ 2x - 2 = 0 ] [ 2x = 2 ] [ x = 1 ]

So, ( x = 1 ) is a critical point.

To determine if it's a maximum, minimum, or neither, you can use the second derivative test. If ( f''(x) > 0 ), it's a minimum; if ( f''(x) < 0 ), it's a maximum. If ( f''(x) = 0 ), the test is inconclusive.

  1. Find the second derivative ( f''(x) ). [ f''(x) = 2 ]

Since ( f''(x) ) is positive for all ( x ), the critical point ( x = 1 ) corresponds to a minimum.

Now, to find the inflection point(s), you need to find where the concavity changes, which occurs where the second derivative ( f''(x) ) equals zero or is undefined. Since ( f''(x) ) is a constant function in this case, there are no inflection points for ( f(x) = x^2 - 2x - 3 ).

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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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