Determine the critical points of f(x) = e^x - 2x. (2021)

₹0.0

Question: Determine the critical points of f(x) = e^x - 2x. (2021)

Options:

Correct Answer: 1

Solution:

f\'(x) = e^x - 2. Setting f\'(x) = 0 gives e^x = 2, so x = ln(2).

Determine the critical points of f(x) = e^x - 2x. (2021)

Determine the critical points of f(x) = e^x - 2x. (2021)

Step 1: Write down the function f(x) = e^x - 2x.
Step 2: Find the derivative of the function, which is f'(x). The derivative of e^x is e^x and the derivative of -2x is -2. So, f'(x) = e^x - 2.
Step 3: Set the derivative equal to zero to find critical points. This means we solve the equation e^x - 2 = 0.
Step 4: Rearrange the equation to isolate e^x. This gives us e^x = 2.
Step 5: To solve for x, take the natural logarithm (ln) of both sides. This gives us x = ln(2).
Step 6: The critical point is x = ln(2).

Differentiation – Understanding how to find the derivative of a function to identify critical points.
Exponential Functions – Recognizing the properties of exponential functions, particularly in solving equations involving e^x.
Natural Logarithm – Applying the natural logarithm to solve for x when dealing with exponential equations.