Calculus- Inflection Pointa

Question

Find the points of inflection and discuss the concavity of the function f(x) = −8x − 2cosx on the interval (0,2π )

Answer 1

Set the second derivative and set it equal to zero.

 

y'= -8 + 2sinx

 

y'' = 2cosx

 

 

Then,

 

2cosx = 0

 

 cos(x) = 0

 

 x = π/2

 x = 3π/2

 

 

These x values are your possible points of inflection.  All you have to do is evaluate the second derivative around these points.

 

If the second derivative changes signs at the x value, then there is a point of inflection at that x value.

If the second derivative is positive, then concave up.

If the second derivative is negative, then concave down.

 

Evaluate these second derivatives in the order given to see where the signs change.

 

f''(π/6) ,  f''(π)  ,  f''(2π)