If g is a differentiable function such that g(x) < 0 for all real numbers x and if f'(x)=(x2-4)g(x), which of the following is true?
The answer is "B", f(x) has a minimum at x=-2 and a maximum at x=+2.
For explanations, see
http://www.jiskha.com/display.cgi?id=1293845056
Since g(x) is a differentiable function and g(x) < 0 for all real numbers x, let's consider the function f(x) = x^2 - 4.
To find the critical points and intervals where f'(x) is positive and negative, we first need to find where f'(x) = 0.
f'(x) = (x^2 - 4)g(x) = 0
Since g(x) ≠ 0 for all x, we can divide both sides of the equation by (x^2 - 4) to get:
x^2 - 4 = 0
This equation factors as (x + 2)(x - 2) = 0.
So, the critical points are x = -2 and x = 2.
Now let's consider the intervals and the sign of f'(x).
For x < -2, both factors (x + 2) and (x - 2) are negative. Therefore, f'(x) = (x^2 - 4)g(x) < 0.
Between -2 and 2, (x + 2) < 0 and (x - 2) > 0. Therefore, f'(x) = (x^2 - 4)g(x) > 0.
For x > 2, both factors (x + 2) and (x - 2) are positive. Therefore, f'(x) = (x^2 - 4)g(x) < 0.
Based on this analysis, we can conclude that f'(x) is positive between -2 and 2, and negative outside of that interval.
Therefore, the correct statement is: f'(x) is positive for -2 < x < 2.
To determine which of the given options is true, we need to analyze the information provided.
We know that g(x) is a differentiable function and g(x) < 0 for all real numbers x. Additionally, we have the derivative of a function f'(x) = (x^2-4)g(x).
Let's consider the given options:
A) f(x) is an increasing function for all x.
B) f(x) is a decreasing function for all x.
C) f(x) has exactly one local maximum.
D) f(x) has exactly one local minimum.
To find the answer, we can use the information from the derivative and the fact that g(x) < 0 for all x.
Looking at the derivative f'(x) = (x^2-4)g(x), we see that for x^2-4 to be negative, x must be between -2 and 2. This implies that the parabola x^2-4 is negative for x values within that range.
Since g(x) < 0 for all x, the product (x^2-4)g(x) will always be positive. This means that the function f(x) will always have a positive derivative, and hence it will always be an increasing function.
Therefore, the correct answer is option A) f(x) is an increasing function for all x.