The function f(x) = -x? + 2x + 6 is graphed in the xy-coordinate plane as shown.
Based on the graph of the function, which of the following statements are true? Select all that apply.
O f is increasing on the interval x < 0.
O f is decreasing on the interval x < 0.
O f is increasing on the interval 0 < x < 1.
O f is decreasing on the interval 0 < x < 1.
O f is increasing on the interval 1 < x < 3.
O f is decreasing on the interval 1 < x < 3.
O f is increasing on the interval x > 3.
O f is decreasing on the interval x > 3.
O f is increasing on the interval 1 < x < 3.
O f is decreasing on the interval x < 0.
O f is increasing on the interval x > 3.
can you explain
Certainly! Let's first look at the function f(x) = -x^2 + 2x + 6.
To determine where the function is increasing or decreasing, we look at the slope of the function. The slope of a function is given by its derivative.
First, we find the derivative of f(x).
f'(x) = -2x + 2
Now, let's analyze the sign of f'(x) to determine where the function is increasing or decreasing.
1. f is increasing on the interval 1 < x < 3:
For 1 < x < 3, the derivative f'(x) = -2x + 2 is positive. This means that the slope of the function is positive in this interval, so f(x) is increasing.
2. f is decreasing on the interval x < 0:
For x < 0, the derivative f'(x) = -2x + 2 is negative. This means that the slope of the function is negative in this interval, so f(x) is decreasing.
3. f is increasing on the interval x > 3:
For x > 3, the derivative f'(x) = -2x + 2 is positive. This means that the slope of the function is positive in this interval, so f(x) is increasing.
Therefore, based on the graph, the statement "f is increasing on the interval 1 < x < 3, x > 3; and f is decreasing on the interval x < 0" is true.