Determine whether the statement is true or false.
There exists a function f such that
f(x) > 0,
f '(x) > 0,
and
f ''(x) < 0
for all x.
true or false question
so, you want f(x) to be positive, increasing, concave down
How about f(x) = √x
Not sure if it's possible with a domain of all reals ...
To determine whether the statement is true or false, we need to consider the properties of the given function f(x) and its derivatives.
Given that f(x) > 0 for all x, we know that the function is always positive.
The first derivative, f'(x), represents the rate of change of the function. The condition f'(x) > 0 means that the function is increasing for all x. This indicates that the slope of the function is positive throughout its domain.
The second derivative, f''(x), represents the rate of change of the first derivative, or the concavity of the function. The condition f''(x) < 0 means that the second derivative is negative. This implies that the slope of the slope, or the concavity, is negative for all x.
To determine if there exists a function that satisfies these conditions, we can look at a simple example. Let's consider the function f(x) = -x^2 + 1.
- f(x) = -x^2 + 1 is always greater than 0 because the quadratic term (-x^2) dominates the constant term (1) and the parabola opens downwards.
- f'(x) = -2x is always greater than 0 because the linear term (-2x) is negative, indicating that the function is monotonically increasing.
- f''(x) = -2 is less than 0, indicating that the function is concave downward.
Therefore, we have found an example function that satisfies all the given conditions. So the statement is true.
In general, to determine the truth of such a statement, we can employ mathematical analysis techniques to find a function that satisfies the given conditions or provide a counterexample that disproves it.