Home > Calculus > Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)

Is #f(x)=-e^(x^2+2) # increasing or decreasing at #x=0 #?

Answer 1

Ezekiel Alexander

Find the first derivative, then evaluate at #x=0#

#f'=-(2x)e^(x^2+2)#

#f'(0)=0#

So, at #x=0#, there is a critical point where the function is neither increasing nor decreasing

hope that helped

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Answer 2

Evelyn Agard

To determine if ( f(x) = -e^{x^2+2} ) is increasing or decreasing at ( x = 0 ), we need to examine the sign of its derivative at that point.

The derivative of ( f(x) ) with respect to ( x ) is given by ( f'(x) = -2xe^{x^2+2} ).

Now, evaluate ( f'(0) ): [ f'(0) = -2(0)e^{0^2+2} = 0 ]

Since ( f'(0) = 0 ), we cannot determine the increasing or decreasing behavior of ( f(x) ) at ( x = 0 ) using the first derivative test. We may need to apply higher-order derivative tests or examine the function's behavior in the surrounding interval to determine its behavior more accurately.

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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.