Final answer:
The given expression represents the profit of a new business. By analyzing the behavior of the profit function, we can determine which statement is true for the given range. The correct statement is that the profit is maximized at x=6.
Step-by-step explanation:
The given expression, 3x³−8x²−240x, represents the profit a new business makes in dollars. To determine which statement is true for the range 0≤x≤12, we need to analyze the behavior of the profit function.
To find if the profit is always positive, we can check if the function is above the x-axis for all x values in the given range. We can also check if the profit is zero at x=0 and if it is negative for all x. Finally, we can investigate if the profit is maximized at x=6.
To do this, we can graph the profit function and analyze its behavior. Alternatively, we can find the critical points of the function by taking its derivative and solving for x. By analyzing the behavior of the function, we can determine the correct statement.
The question concerns the profitability of a business in relation to the number of goods or services sold (x) within a certain range (0≤x≤12). Analyzing the profit function 3x³−8x²−240x, we can determine the nature of profit over the given range.
To see if the profit is zero at x=0, we substitute x with 0 and get 3(0)³−8(0)²−240(0)=0, so choice (b) 'The profit is zero at x=0' is correct.
To examine the rest of the statements, we need to consider the structure of the function, which is a cubic polynomial.
Based on the analysis, the correct statement is d) The profit is maximized at x=6.