Question

F is increasing on the interval (-2,3); f is decreasing on the intervals (-infinity, -2) and (3, infinity). f(x) >0 on the intervals (-infinity, -4) and 1,5). f(x) <0 on the intervals (-4,1) and (5,infinity).

Options:
a) True
b) False

Answer 1

The function f(x) can correspond to y = x² based on the given information.

Step-by-step explanation:

The given information states that the function f is increasing on the interval (-2,3) and decreasing on the intervals (-∞, -2) and (3, ∞). It also states that f(x) > 0 on the intervals (-∞, -4) and (1,5), and f(x) < 0 on the intervals (-4,1) and (5,∞).

We need to determine which option, y = 13x or y = x², could correspond to the function f(x) based on the given information.

Let’s analyze the options:

a) y = 13x:

This option represents a linear function with a slope of 13. This means that the function is increasing with a positive slope. However, this option does not match the information that f is decreasing on the intervals (-∞, -2) and (3, ∞).

b) y = x²:

This option represents a quadratic function with a positive coefficient for x². This means that the function is increasing on the interval (0,∞). Additionally, since the coefficient is positive, the function is concave up, indicating that it is also increasing on the interval (-∞,0). This matches the information provided about f being increasing on the interval (-2,3).

Therefore, the correct option that could correspond to f(x) based on the given information is b) y = x².