Locating critical points Find the critical points of the follo...

Question

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = eˣ + e⁻ˣ

Accepted Answer

Welcome back, everyone. Determine the critical points of the function F of X equals 4 e's the power of X plus 4Es the power of negative X A 0.0 and 0.8. B 0.8. C 0.4, and D 0.0 and 0.4. If we want to identify the critical points of a function, well, we have to differentiate it. So we have to find the derivative of 4 e to the power of X plus 4 e to the power of negative X. We can factor out the constants. And we're going to get 4 E to the power of X. Plus 4Es the power of negative X multiplied by the derivative of negative X based on the chain rule, right? So in this context, we're going to get 4 E to the power of X minus because the derivative of negative X is -1. 4 eats the power of negative x and we can rewrite it as 4Es the power of x minus 4 divided by e to the power of X, and now we're going to find the common denominator which is e to the power of x. So we get 4 e to the power of 2 X because we have to multiply e to the power of x by itself, so we're squaring it, and then we are subtracting 4. So now we have to check two cases. First of all, when does the derivative not exist? Well, it exists for all X access, right? Because the denominator is never equal to zero since we have an exponential function. So each the power of X is greater than 0 for all the X values, meaning the only case that we want to check is when the derivative is equal to 0, and because we have the irrational expression, we simply want to set. Or it's the power of 2 x minus 4 equal to 0. Now, what we're going to do is basically factor it out, we're going to get 4 E, it's the power of 2 X minus 1 is equal to 0. And now we're going to set E to the power of 2X minus 1 is equal to 0, which essentially gives us E to the power of 2 X. This must be equal to 1. Now, when is this the case? Well, essentially this is the case when our exponent is 0, so X must be equal to 0. If we're looking for the vertical points, we have the X coordinate. We now want to identify the Y coordinate. So let's go ahead and solve for F of 0. That would be 4 E to the power of 0, plus 4e the power of negative 0. Well, basically that's 4 + 4. Right, because we can say 0 which is 0 and we get 8. So the critical point would be X equals 0, Y equals 8, which essentially corresponds to the answer choice B. Thank you for watching.

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = eˣ + e⁻ˣ

Key Concepts

Critical Points

Derivative

Exponential Functions

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