Find the minimum value of the function, f(x) = x*exp(x)

The minimum value lies where the tangent to the curve has a gradient of zero and the curve approaching from both directions increases in value. This is done by finding the first and second derivatives of the function. df/fx = xexp(x)+exp(x) Set this equal to zero and solve for x: xexp(x)+exp(x)=0 exp(x) * (x+1)=0 The solution lies in one of the expressions exp(x) or (x+1) being equal to zero.exp(x)=0 has no solution, therefore only 1 solution when (x+1)=0, which is x=-1. We can check our solution is a minimum as d2f/dx2 > 0 for a minimum: d2f/dx2 = x*exp(x) + 2exp(x) @ x=-1 d2f/dx2= 0.368 hence a minimum. Finally, the value of the function at x=-1 is given by the function, f=-0.368