Describe the concavity upward and downward

Question

#1

 

f(x) = x [Times the SQUARE ROOT OF(x+24)]

 

 

 

 

 

 

 

 

 

 

 

 

#2

 

f(x) = x [Times the SQUARE ROOT OF(12-x)]

 

 

Answer 1

#1.  f(x) = x(x+24)^½

  f' = (x+24)^½ + x(½)(x+24)^-½

f" = [¾•x+24] / (x+24)^-3/2

 

The only key point (zero of either numerator or denominator) is x=-32 but that's not in the domain of this function.

Therefore f" is always positive and thus always concave upward.  Do you agree?