When using the first derivative test to find the critical points of a function, do you always have to include #x=0?#

Answer 1

Yes, if #0# is in the function's domain. Note that if the derivative isn't defined at #0#, that is also a critical point.

That is because #0# can be a critical point of a function. For example, consider the function #f(x) = x^2#.
#f'(x) = 2x#, and the root to the equation #f'(x) = 0 => 2x = 0# is
#x = 0#.
At #0#, our #f# also happens to have an absolute minimum, which wouldn't always be the case.
In fact, many such situations exist, so, unless #0# is not in the functions domain, there is no reason why you wouldn't check what happens there.
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Answer 2

No, you don't always have to include ( x = 0 ) when using the first derivative test to find critical points of a function. Critical points occur where the first derivative is either zero or undefined. So, you only need to include ( x = 0 ) if it satisfies the conditions for a critical point according to the first derivative test (i.e., if the derivative is zero or undefined at ( x = 0 )). If the first derivative is not zero or undefined at ( x = 0 ), then it's not considered a critical point.

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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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