# How to find the intervals on which the following function is a)increasing b)decreasing c)concave up d)concave down.

y = 2x, x < 1

2 - x^2, x >= 1

Thank you.

## To find the intervals on which a function is increasing or decreasing, we need to determine where the derivative of the function is positive or negative. Then, to find the intervals of concavity, we need to determine where the second derivative is positive or negative.

First, let's find the derivative of y with respect to x.

For x < 1, y = 2x. The derivative of y with respect to x is simply 2.

For x >= 1, y = 2 - x^2. To find the derivative, we need to use the power rule, which states that if y = x^n, then dy/dx = nx^(n-1). Taking the derivative of (2 - x^2) with respect to x, we get -2x.

Now, let's find the second derivative of y.

For x < 1, the derivative of y with respect to x is 2, and the second derivative is 0.

For x >= 1, the derivative of y with respect to x is -2x, and the second derivative is -2.

Now, let's analyze the intervals:

a) To find the intervals on which the function is increasing, we look for where the derivative is positive.

For x < 1, the derivative is positive (2) for all values of x.

For x >= 1, the derivative is negative (-2x) for all values of x.

Therefore, the function is increasing for all values of x < 1 and decreasing for all values of x >= 1.

b) To find the intervals on which the function is decreasing, we look for where the derivative is negative.

For x < 1, the derivative is positive (2) for all values of x.

For x >= 1, the derivative is negative (-2x) for all values of x.

Therefore, the function is decreasing for all values of x < 1 and increasing for all values of x >= 1.

c) To find the intervals of concavity up, we look for where the second derivative is positive.

For x < 1, the second derivative is 0.

For x >= 1, the second derivative is negative (-2).

Therefore, the function is concave down for all values of x < 1 and concave up for all values of x >= 1.

d) To find the intervals of concavity down, we look for where the second derivative is negative.

For x < 1, the second derivative is 0.

For x >= 1, the second derivative is negative (-2).

Therefore, the function is concave up for all values of x < 1 and concave down for all values of x >= 1.

In summary:

a) The function is increasing for all values of x < 1.

b) The function is decreasing for all values of x >= 1.

c) The function is concave up for all values of x >= 1.

d) The function is concave down for all values of x < 1.