What are the intervals of increasing and decreasing, and the intervals of concavity of the following?

Question

What are the intervals of increasing and decreasing, the local min and max and the intervals of concavity of the following?

 

y = e^xsin(x)  -pi ≤ x ≤ pi

 

I know I need the first derivative, which is e^x(sin(x) + cos(x)) and then set it = 0. However, I'm not sure how to solve for x at this point.

 

 

ANSWER: Increasing on (pi/4, 3pi/4); decreasing on  (-pi,-pi/4) (3pi/4, pi)

Concave up on (-pi/2, pi/2); concave down on (-pi, -pi/2), (pi/2, pi)

#29, p352

Answer 1

e^x(sinx + cosx) = 0 when either of the following is true:

 

e^x=0 OR (sinx + cosx) = 0.

 

e^x can never equal zero. 

 

When does sinx + cosx = 0 in the interval from -pi to +pi?

 

One way to solve this is to divide every term of the equation by cosx (for example).

 

tanx + 1 = 0

 

tanx = -1

 

x = arc tan -1

 

In the interval from -pi to pi this happens at reference angles of 45 degrees or pi/4 where tangent is negative.

 

That would be 2nd and 4th quadrants at -pi/4 and 3pi/4.

 

Those are your critical numbers. Examine the sign of the 1st derivative in the intervals (-pi, -pi/4), (-pi/4, 3pi/4), and (3pi/4, pi). When positive the original function is increasing, when negative decreasing.

 

You will need to find 2nd derivative and set equal to zero to determine point(s) of inflection and concavity.

 

See: https://www.desmos.com/calculator/7kgqlz9dap