Question

For the function f(x)=x2+11, we need to determine the interval on which the function is increasing. To do this, you should find the derivative f′(x) and then identify where it is positive. In this case, the function is always increasing, so the largest on interval is the entire real line.

A) 1, [infinity]
B) -[infinity] , 1
C) -[infinity] , 0
D) 0, [infinity]

Answer 1

The function f(x)=x^2+11 is always increasing, therefore it increases on the interval (-∞, ∞). Continuous probability functions define probability as the area under the curve, with the total area equalling 1 for the entire function.

Step-by-step explanation:

The student is seeking help in finding the interval on which the function f(x)=x^2+11 is increasing. To determine this, we calculate the first derivative f'(x), which is simply 2x. Since this derivative is positive for all values of x greater than 0, the function is always increasing for x > 0. However, as the derivative is 0 at x = 0 and positive again for x < 0, we can conclude that the function is actually increasing for all values of x. Therefore, the function is increasing on the interval (-∞, ∞), which represents the entire real line.

When we talk about continuous probability functions, the area under the curve represents the probability. For non-negative functions that represent probability distributions, the total area under the curve must equal 1 since the probability cannot exceed 100%. In such cases, the probability is defined as the area under the curve between two points on the x-axis. Considering continuous probability functions, intervals outside the defined range, like P(x > 15) when the function is defined between 0 ≤ x ≤ 15, will have a probability of 0.