Question

How to find critical points of eˣ(cos(x)−sin(x)).

Answer 1

To find the critical points of the function eˣ(cos(x)-sin(x)), calculate the derivative of the function and set it equal to zero. The critical points occur when sin(x) is equal to zero, so the critical points are x = nπ, where n is an integer.

Step-by-step explanation:

To find the critical points of the function eˣ(cos(x)-sin(x)), we need to find the points where the derivative of the function equals zero. Let's start by finding the derivative of the function using the product rule and chain rule:

f'(x) = (eˣ)(-sin(x)-cos(x)) + (eˣ)(cos(x)-sin(x))

Simplifying the derivative:
f'(x) = -2eˣsin(x)

Now, set the derivative equal to zero and solve for x:
-2eˣsin(x) = 0

eˣsin(x) = 0

Since eˣ is always positive, we only need to consider when sin(x) = 0. The critical points occur when x = nπ, where n is an integer.

So, the critical points of the function are x = nπ, where n is an integer.