(a) To find the profit function, we need to subtract the cost function from the revenue function.
Profit = Revenue - Cost
So, we need to subtract C(x) from R(x).
P(x) = R(x) - C(x)
P(x) = (-0.6x^2 + 19x) - (0.4x^2 - 29x + 44)
P(x) = -0.6x^2 + 19x - 0.4x^2 + 29x - 44
P(x) = -1x^2 + 48x - 44
Therefore, the profit function is P(x) = -x^2 + 48x - 44.
(b) To find the vertex of the profit function, we can use the formula x = -b/2a, where the profit function is in the form ax^2 + bx + c.
In this case, a = -1 and b = 48.
x = -b/2a
x = -(48) / (2 * (-1))
x = -48 / -2
x = 24
Now we need to find the corresponding y-value by substituting x = 24 into the profit function.
P(24) = -(24)^2 + 48(24) - 44
P(24) = -576 + 1152 - 44
P(24) = 532
The vertex of the profit function is (24, 532).
In this context, the vertex represents the level of production (x) that maximizes the profit (P). So, when the company produces 24 hundred pencils, it maximizes its profit, which is equal to $532,000.