Question 60372
You can plot this equation or just picture what it should look like
I'll just try to describe it.
(1) Because the {{{w^2}}} term is positive, the graph is a parabola
that slopes negatively down from the left, comes down to a minimum and
slopes back up positively.
(2) The problem is to find the minimum. The minimum will be half way
between two values of w that have the same C(w), in other words, half
way between the points (w[1],C[1]) and (w[2],C[2]) where C[1] = C[2]
{{{C = .04*w^2 - 6*w + 275}}}
If they don't produce any widgets, the fixed cost is $275
{{{C = .04*0 - 6*0 + 275}}}
{{{C = 275}}}
Now, if I find another value of w for which C(w) = 275, the C(w) mid-
way between (0,275) and (w[2],275) will be minimum cost
{{{C = .04*w^2 – 6*w + 275}}}
{{{275 = .04*w^2 – 6*w + 275}}}
subtract 275 from both sides
{{{.04*w^2 – 6*w = 0}}}
{{{w*(.04*w - 6) = 0}}}
If the 1st factor, w, is zero, that's the point we already have, (0,275)
so, set the 2nd factor equal to zero.
{{{.04*w - 6 = 0}}}
{{{.04*w = 6}}}
{{{w = 150}}}
That's our 2nd point. We now have (0,275 and (150,275)
The minimum C is halfway between the w's, or {{{150/2 = 75}}}
That's the answer to (a), 75 widgets made give minimum cost
Now find C(w)
{{{C = .04*w^2 - 6*w + 275}}}
{{{C = .04*(75)^2 - 6*(75) + 275}}}  
{{{C = .04*5625 - 450 + 275}}}
{{{C = 225 - 450 + 275}}}
{{{C = 50}}}
That means the min is at (75,50) and the answer to (b) is
the minimum unit cost is $50
A simple test will tell you if this is truly the min
Find C(w) for w = 74.9 and for w= 75.1. These values of w should 
give costs that are both slightly higher than $50.
I get C(w) = 50.0004 for both