Calculus- Concavity!

Question

Determine the open intervals on which the graph of the function y = 3x − tanx, (-π/2, π/2i) s concave upward or concave downward.

Answer 1

Set the second derivative of y and set it equal to zero.

 

y' = 3 - sec²(x)

 

y'' = -2sec(x)sec(x)tan(x)

 

 

Therefore,

 

-2sec²(x)tan(x) = 0

 

-2(1/cos²(x))(sin(x)/cos(x)) = 0

 

-2sin(x) / cos³(x) = 0

 

 

Set    sin(x)=0   to find the points of inflection, keeping in mind that  cos(x)≠0.

 

If the second derivative around the point of inflection is positive, then concave up.

If the second derivative around the point of inflection is negative, then concave down.