# What are the critical points of #f (x) = e^x + e^-(6x)#?

This is a critical value.

graph{e^x+3^(-6x) [-10.04, 9.96, -5.04, 4.96]}

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To find the critical points of ( f(x) = e^x + e^{-6x} ), you need to first find its derivative and then solve for where the derivative equals zero.

The derivative of ( f(x) ) with respect to ( x ) is:

[ f'(x) = e^x - 6e^{-6x} ]

To find critical points, set ( f'(x) ) equal to zero and solve for ( x ):

[ e^x - 6e^{-6x} = 0 ]

[ e^x = 6e^{-6x} ]

[ e^x = \frac{6}{e^{6x}} ]

[ e^{7x} = 6 ]

[ x = \frac{1}{7} \ln(6) ]

Therefore, the critical point of ( f(x) ) is ( x = \frac{1}{7} \ln(6) ).

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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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