HW Helper P: 1,025 If you take the first order partials, set them = 0 , and solve the resulting system, you get: $$x = 2, y = 3, z = \frac{10}{3}, t = \frac{7}{3}, w = \pi$$ but these values do not fit the first constraint, namely that $$0\leq \frac{x}{2}+y+3z+3y+\frac{5}{2}w< 30$$ in fact those values of x,y,z,t, and w, will give $\frac{x}{2}+y+3z+3y+\frac{5}{2}w=23+\frac{5}{2}\pi = 30.86 \mbox{ (appox.)}$ Also note that $f(1,1,6.70000000000000016,1,1)=-\infty$.
 HW Helper P: 1,025 Crap, I minimized. Let me look again. Well, you might try Lagrange Multipliers with the constraints as equalities for the upper and lower bounds of the inequalities. That version of Lagrange Multipliers goes: To find the extrema of $f(x,y,z,...)$ subject to the constraints $g_{1}(x,y,z,...)=k_{1}, g_{2}(x,y,z,...)=k_{2}, g_{3}(x,y,z,...)=k_{3},...$ set $$\vec{\nabla} f(x,y,z,...) = \lambda_{1}\vec{\nabla} g_{1}(x,y,z,...) + \lambda_{2}\vec{\nabla} g_{2}(x,y,z,...) + \lambda_{3}\vec{\nabla} g_{3}(x,y,z,...) +\cdot\cdot\cdot$$ and solve as normal.