Strictly Convex Function

Convex Decreasing Function

Convexity of the Cost Curve Implies Increasing Marginal Cost

A company models its total production expense using the function $E(q) = 0.1q^3 - 3q^2 + 50q + 1000$, where $q$ is the quantity of units produced ($q \ge 0$). For which range of production quantities is the rate of change of the expense non-decreasing?

A firm's total cost of production is modeled by the function $C(q) = q^3 - 6q^2 + 20q + 50$, where $q$ is the quantity of units produced ($q \ge 0$). An analyst states that the cost function is convex for all quantities $q > 2$. Which of the following statements provides the correct reasoning to support the analyst's claim?

If a twice-differentiable function's slope is always positive over its entire domain, the function must be convex.

Analyzing an Experimental Outcome

The graph below represents the first derivative, $f'(x)$, of a function $f(x)$. The graph of $f'(x)$ is a parabola that opens upwards and has its minimum point (vertex) at $(2, -4)$. On which interval is the function $f(x)$ convex?

Connecting Geometric and Calculus Views of Convexity

A company, 'TechForward', develops a revolutionary manufacturing process that allows it to produce a high-demand electronic component for $20 per unit. All of its competitors produce the same component at a cost of $50 per unit and sell it for $70. TechForward has the capacity to meet a large portion, but not all, of the current market demand. To maximize its profits from this cost advantage, what pricing strategy should TechForward adopt?

A firm is evaluating potential mathematical models for its total cost function, $C(q)$, where $q$ is the quantity of goods produced ($q \ge 0$). The firm's production process is known to exhibit a non-decreasing marginal cost. Which of the following functions correctly models this economic characteristic for all $q \ge 0$?

A firm's production manager observes that for every additional unit produced, the cost of producing that specific unit is higher than the cost of producing the previous one. This is true across all production levels. If the firm's total cost is represented by a smooth, twice-differentiable function of the quantity produced, which of the following mathematical properties must this function exhibit to accurately model the manager's observation?

For each function provided, match it with the description that correctly characterizes the behavior of its slope over its entire domain.