A differentiable function called f(x) achieves its maximum when x=0. Which of the following must then be true?
1. The function p(x) = xf(x) has a critical point when x = 0.
2. The function m(x) = (f(x))^2 has its maximum at x = 0.
3. The function j(x) = f(x^2) has its maximum at x = 0.
#1. No
consider f(x) = e^(-x^2)
#2. No
consider f(x) = -x^2
See what you can do with #3.
To determine which of the given statements must be true, we can analyze each one individually:
1. The function p(x) = xf(x) has a critical point when x = 0.
To verify this statement, we need to find the critical points of p(x). The critical points occur where the derivative of p(x) equals zero or is undefined. Taking the derivative of p(x), we get:
p'(x) = f(x) + x * f'(x).
Setting p'(x) equal to zero and solving for x, we have:
0 = f(x) + x * f'(x).
Since f(x) achieves its maximum when x = 0, we know that f'(x) = 0 at x = 0. Therefore, plugging in x = 0 into the above equation, we get:
0 = f(0).
Since f(x) is differentiable and achieves its maximum at x = 0, we conclude that f'(0) = 0. Hence, p'(0) = f(0) + 0 * 0 = f(0) = 0.
Therefore, the function p(x) = xf(x) does indeed have a critical point at x = 0. Hence, statement 1 is true.
2. The function m(x) = (f(x))^2 has its maximum at x = 0.
We already know that f(x) achieves its maximum at x = 0. So, to check statement 2, we need to analyze the function m(x) = (f(x))^2. Taking the derivative of m(x) with respect to x, we get:
m'(x) = 2f(x)f'(x).
For m(x) to have its maximum at x = 0, we need m'(0) = 0. However, in general, m'(0) = 2f(0)f'(0), and since f'(0) = 0, we have m'(0) = 0.
Therefore, the function m(x) = (f(x))^2 does have its maximum at x = 0. Hence, statement 2 is also true.
3. The function j(x) = f(x^2) has its maximum at x = 0.
To verify this statement, we need to analyze the function j(x) = f(x^2). The maximum point of j(x) depends on how the function f(x^2) behaves.
However, based on the given information, it is not possible to conclude that j(x) has its maximum at x = 0. The maximum of f(x^2) depends on the function itself and how it behaves for positive and negative values of x.
Therefore, we cannot determine if statement 3 is true based on the given information.
In summary, statements 1 and 2 are true, while statement 3 cannot be determined.
To determine which of the statements are true, we need to analyze the given information about f(x) achieving its maximum at x = 0.
1. The function p(x) = xf(x) has a critical point when x = 0.
To check if p(x) has a critical point at x = 0, we need to find the derivative of p(x) and see if it equals zero at x = 0.
First, let's find p'(x), the derivative of p(x):
p'(x) = (xf(x))' = f(x) + x(f(x))'
Now, let's substitute x = 0 to see if we have a critical point:
p'(0) = f(0) + 0(f(0))'
Since f(x) achieves its maximum at x = 0, the derivative f'(0) = 0, so f(0)' = 0. Therefore, p'(0) = f(0) + 0 = f(0), which means p(x) does have a critical point at x = 0.
Hence, statement 1 is true.
2. The function m(x) = (f(x))^2 has its maximum at x = 0.
To determine if m(x) has its maximum at x = 0, we need to evaluate the second derivative of m(x) and see if it is negative at x = 0.
First, let's find m''(x), the second derivative of m(x):
m''(x) = [2(f(x))f'(x)]'
Now, let's substitute x = 0 to check if it is negative:
m''(0) = [2(f(0))f'(0)]' = [2(f(0))(0)]' = 0
Since the second derivative m''(0) = 0, we cannot determine whether m(x) has its maximum at x = 0 based on the given information. So, statement 2 may or may not be true.
3. The function j(x) = f(x^2) has its maximum at x = 0.
To determine if j(x) has its maximum at x = 0, we need to analyze the behavior of f(x^2) around x = 0.
Since f(x) achieves its maximum at x = 0, we know that f'(0) = 0. To determine whether j(x) has its maximum at x = 0, we need to consider the composition of functions and analyze the behavior of f(x^2) near x = 0.
Since x^2 approaches 0 as x approaches 0, we can consider the behavior of f(x^2) as x approaches 0.
If f(x^2) has its maximum at x = 0, it means that f(x^2) is increasing on one side of x = 0 and decreasing on the other side.
However, without further information about the function f(x), we cannot determine the behavior of f(x^2) around x = 0. Therefore, statement 3 may or may not be true.
In conclusion:
Statement 1 is true.
Statement 2 may or may not be true.
Statement 3 may or may not be true.