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### Problem 5: Productivity and Labour
Let \( x = f(L) \) be the output when \( L \) units of labor are used as input. Given that:
- \( f(0) = 0 \)
- \( f'(L) > 0 \) for all \( L > 0 \) (indicating that output increases with more labor)
- \( f''(L) < 0 \) for all \( L > 0 \) (diminishing returns to labor)

The **average productivity** is defined by \( g(L) = \frac{f(L)}{L} \).

#### (a) Indicate on a figure the values of \( f'(L^*) \) and \( g(L^*) \). Which is larger?
- \( f'(L^*) \) is the marginal productivity (the derivative of output with respect to labor).
- \( g(L^*) \) is the average productivity.

From the conditions \( f'(L) > 0 \) and \( f''(L) < 0 \), the marginal productivity is decreasing but still positive, and the average productivity could initially increase and then decrease. Typically, when labor input is small, marginal productivity \( f'(L^*) \) is higher than average productivity \( g(L^*) \), but as labor increases, average productivity can surpass marginal productivity.

#### (b) How does the average productivity change when labor input increases?
Given that \( f'(L) > 0 \) and \( f''(L) < 0 \), the function \( g(L) = \frac{f(L)}{L} \) will increase initially but eventually decrease. This suggests that average productivity first rises with labor but eventually falls due to diminishing returns.

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### Problem 6: Intervals of Increase for Functions
For each of the following functions, determine the intervals where the function is increasing.

#### (a) \( y = 3x^2 - 12x + 13 \)
Find the derivative:
\[
y' = 6x - 12
\]
Set \( y' > 0 \):
\[
6x - 12 > 0 \quad \Rightarrow \quad x > 2
\]
Thus, the function is increasing for \( x > 2 \).

#### (b) \( y = \frac{1}{4}(x^4 - 6x^2) \)
Find the derivative:
\[
y' = \frac{1}{4}(4x^3 - 12x) = x^3 - 3x
\]
Set \( y' > 0 \):
\[
x(x^2 - 3) > 0
\]
The critical points are \( x = -\sqrt{3}, 0, \sqrt{3} \). Use a sign chart to determine

Let x = f(L) be the output when L units of labour are used as input. Assume that f(0) = 0 and that f'(L) > 0, f''(L) < 0 for all L > 0. Average productivity is defined by the formula g(L) = f(L)/L. How do we analyze the average and marginal productivity and find intervals of increase for given functions?

Analyze average and marginal productivity using differentiation. This includes finding g(L) = f(L)/L, interpreting diminishing returns, and identifying intervals of increase for various functions. Ideal for undergraduate economics and calculus courses.

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