Question

When comparing the f(x) = –x2 + 2x and g(x) = log(2x + 1), on which interval are both functions positive?

(–∞, 0)
(0, 2)
(2, ∞)
(∞, ∞)

Answer 1

Both functions f(x) = −x^2 + 2x and g(x) = log(2x + 1) are positive on the interval (0, 2). This is derived by analyzing each function's individual intervals of positivity and finding their intersection.

Step-by-step explanation:

To find on which interval both f(x) = −x^2 + 2x and g(x) = log(2x + 1) are positive, we first need to determine the intervals on which each individual function is positive.

For the quadratic function f(x), setting the function positive gives:

• f(x) > 0
• −x^2 + 2x > 0
• x(−x + 2) > 0

This quadratic has roots at x = 0 and x = 2. Since it opens downward (the coefficient of x^2 is negative), the function is positive between its roots, which is the interval (0, 2).

For the logarithmic function g(x), the argument of the log function must be greater than zero for the function value to be positive:

• log(2x + 1) > 0
• 2x + 1 > 1 (since log(1) = 0)
• x > 0

Therefore, g(x) is positive for x > 0. Combining both conditions, the interval on which both functions are positive is the intersection of the intervals where they are individually positive, which is (0, 2).