Summary of curve sketching (calculus 1) -
sketch the graph: xtanx, -pi/2 < x < pi/2

  to find the critical points, we need to find the first derivative and equal it to zero as the following : 

y = x * tan(x) 

y ' = x * sec^2(x) + tan(x) * 1 

y ' = ( x / cos^2(x) ) + ( sin(x) / cos(x) ) ---> LCD 

y ' = ( x / cos^2(x) ) + ( sin(x)*cos(x) / cos(x)*cos(x) ) 
y ' = ( x / cos^2(x) ) + ( sin(x)*cos(x) / cos^2(x) ) 
y ' = ( x + sin(x)*cos(x) ) / cos^2(x) 
0 = sec^2(x) * ( x + sin(x) * cos(x) ) 

0 = sec^2(x) > no go 

x + sin(x) * cos(x) = 0 > can't be solve algebraically, so we are going to use a calculator. 

the only solution point is 0, which is the critical point. 

to find the interval of concavities, we need to find the points of inflection by finding the second derivative and equal it to zero as the following: 

y '' = x * 2sec(x)sec(x)tan(x) + sec^2(x) * 1 + sec^2(x) 
y '' = 2x * sec^2(x)tan(x) + 2sec^2(x) 

0 = 2x * sec^2(x)tan(x) + 2sec^2(x) 
0 = 2sec^2(x) * [ x * tan(x) + 1 ] 
0 = sec^2(x) * [ x * tan(x) + 1 ] 

0 = sec^2(x) ---> no go ( this will a vertical asymptote ) which is π/2 

x * tan(x) + 1 = 0 > can't be done algebraically, we are going to use a calculator 

solutions are which are the points of inflection: 

±18.7964043662102 
±15.6441283703330 
±12.4864543952238 
±9.31786646179107 
±6.12125046689807 
±2.79838604578389 

we need to check before and after each number into the second derived equation, so i will show an example as 2.79838604578389 

y '' = 2x * sec^2(x)tan(x) + 2sec^2(x) > when x = 2 
y '' = 2*2 * sec^2(2)tan(2) + 2sec^2(2) 
y '' ≈ -39 ( negative means concaving down ) 

concave down: ( π/2 , 2.79838604578389 ) 

we can categorize each number, if you have a limit to the interval, it will be great. 

The ratio of their perimeters would also be 5:9.

To prove this assume one triangle has lengths 5x, 5x, and 5x. That means the other triangle would have 9x, 9x, and 9x. If you add up the lengths of the first triangle you would get 15x and if you add up the lengths of the second triangle you would get 27x. Let's put that in ratio form:

15x : 27x
divide both sides by x
15:27
divide both sides by 3

5:9