# What are the values and types of the critical points, if any, of #f(x)=3e^(-2x^2))#?

We can find critical points by equating the first derivative to zero:

graph{3e^(-2x^2) [-10, 10, -5, 5]}

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To find the critical points of ( f(x) = 3e^{-2x^2} ), we first need to find its derivative, ( f'(x) ), and then solve for ( x ) when ( f'(x) = 0 ).

[ f'(x) = \frac{d}{dx}(3e^{-2x^2}) = -12xe^{-2x^2} ]

Setting ( f'(x) ) equal to zero and solving for ( x ) gives us the critical points.

[ -12xe^{-2x^2} = 0 ]

[ -12x = 0 ]

[ x = 0 ]

So, the only critical point of ( f(x) ) is ( x = 0 ).

To determine the nature of this critical point, we can use the second derivative test.

[ f''(x) = \frac{d^2}{dx^2}(3e^{-2x^2}) = (-12e^{-2x^2}) - (4x)(-12x)e^{-2x^2} = (24x^2 - 12)e^{-2x^2} ]

At ( x = 0 ), we have:

[ f''(0) = (24(0)^2 - 12)e^{-2(0)^2} = -12 ]

Since the second derivative is negative at ( x = 0 ), the critical point ( x = 0 ) is a maximum point.

So, the critical point ( x = 0 ) of ( f(x) ) is a relative maximum, and the corresponding function value is ( f(0) = 3 ).

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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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