Question

The graph of the function f(x) = (x-1)^3+ 1 is shown which of the following is true? 1. F is increasing on the entire number line 2. F”(x) <0 on the interval (-infinity,1). 3. F”(x)>0 on the interval (1,infinity).

Answer 1

The true statement about the function f is that 3. it is increasing on the interval (1,infinity).

How to determine the true statement about the function

From the question, we have the following parameters that can be used in our computation:

f(x) = (x - 1)³ + 1

To determine the intervals where f' is positive or negative, we need to find the second derivative of f

So, we have

f'(x) = 3(x - 1)²

f''(x) = 6(x - 1)

f''(x) = 0 for x = 1.

Since f'' is a polynomial, it's defined for all real numbers.

So, we have

x = 1

We can analyze the intervals around x = 1 using the sign of f''(x).

For x < 1, f''(x) < 0, which means f' is decreasing.

For x > 1, f''(x) > 0, which means f' is increasing.

So, f is increasing on the interval (1,infinity).