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Which one of these functions is decreasing for all real values of x? Circle your answer - AQA - A-Level Maths: Pure - Question 1 - 2020 - Paper 2
Question 1
Which one of these functions is decreasing for all real values of x? Circle your answer. $f(x) = e^x$ $f(x) = -e^{1-x}$ $f(x) = -e^{-x}$ $f(x) = -e^{-x}$
Worked Solution & Example Answer:Which one of these functions is decreasing for all real values of x? Circle your answer - AQA - A-Level Maths: Pure - Question 1 - 2020 - Paper 2
Step 1
Evaluate the functions for monotonicity
Answer
To determine which function is decreasing for all values of , we can analyze the derivatives of each function.
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For the first function, , the derivative is , which is always positive. Thus, this function is increasing.
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For the second function, , the derivative is: f'(x) = -(-e^{1-x}) rac{d}{dx}(1-x) = e^{1-x}
This derivative is positive, indicating that this function is increasing. -
For the third function, , the derivative is:
This function's derivative is also positive, so it is increasing as well. -
For the fourth function, which is , the analysis remains the same:
As with the third function, it is increasing.
Upon examining the second function, , we realize that for , it approaches , which maintains a negative slope and thus exhibits decreasing behavior overall. Hence, the correct answer is .