The loading distribution, w, on a beam is given by w = -36x^2 + 50x

Question

The loading distribution, w, on a beam is given by w = -36x^2 + 50x where x is the distance in meter from one end of the beam. find the value of x at which maximum loading occurs using the second derivative test.

Answer 1

Maybe the question should be rewritten as:

 

The loading distribution, w, on a beam is given by w = -36x^2 + 50x where x is the distance in meter from one end of the beam. find the value of x  (at which maximum loading occurs using the second derivative test of the equation).

 

The graph is convex but X is not the max point.  X is one of the roots. The other one is (0,0)

 

Try the quadratic equation where a=-36; b=50; c=0

Answer 2

w = -36x² + 50x

 

To find the extreme points, take the first derivative of w wrt x, set it to zero, and solve for x:

 

     dw/dx = -72x + 50

     0 = -72x + 50

     72x = 50

     x = 50/72 = 25/36 meters

 

To test if it's a max or min, take the second derivative.  If the second derivative is negative it's a maximum; if it's positive, it's a minimum:

 

     d²w/dx² = -72

 

So the extreme point at x = 25/36 meters is a maximum.