Final answer:
The functions that are increasing throughout their domains are A. f(x) = x²+2 and D. f(x) = x³+2.
Step-by-step explanation:
To determine which functions are increasing throughout their domains, we need to find the first derivative of each function and determine if it is always positive. If the first derivative is always positive, then the function is increasing. Let's go through each function:
A. f(x) = x²+2
First derivative: f'(x) = 2x
This function is increasing throughout its domain because the first derivative, 2x, is always positive for any value of x.
B. f(x) = -4x
First derivative: f'(x) = -4
This function is not increasing throughout its domain because the first derivative, -4, is always negative and not positive.
C. f(x) = -x³
First derivative: f'(x) = -3x²
This function is not increasing throughout its domain because the first derivative, -3x², is negative for negative values of x and positive for positive values of x.
D. f(x) = x³+2
First derivative: f'(x) = 3x²
This function is increasing throughout its domain because the first derivative, 3x², is always positive for any value of x.
E. f(x) = 3x + 4
First derivative: f'(x) = 3
This function is not increasing throughout its domain because the first derivative, 3, is not positive and not negative. It is a constant.
So, the functions that are increasing throughout their domains are A. f(x) = x²+2 and D. f(x) = x³+2.