91Ó°ÊÓ

In calculus, derivatives are a fundamental tool to understand how a function behaves. They tell us the rate at which a function is changing at any given point. For example, if you have a function representing the distance a car travels over time, its derivative would represent the car's speed at each moment.

In this exercise, we have the first derivative, \(f'(x)\), which shows how the function \(f(x)\) changes as \(x\) increases. At \(x = 5\), \(f'(5) = 2\), meaning the function's rate of change is 2 units per unit change in \(x\). This tells us that at exactly \(x = 5\), the function is growing. However, since the second derivative is negative, this rate of grow slows down as \(x\) increases.

Concavity is about the curvature of a function's graph. It describes how a function bends or curves. A function is concave up if it forms a cup-like shape, meaning it bends upwards, and it's concave down if it forms a cap-like shape, bending downwards.

In our specific problem, we have information indicating that the second derivative, \(f''(x)\), is less than zero for \(x \geq 5\). This negative second derivative implies the function is concave down from \(x = 5\) onwards.

• Concave down means the slope of the tangent line is decreasing.
• Even if the function still increases, it does so at a reducing pace.

Thus, as \(x\) moves from 5 to 7, \(f'(x)\) decreases which indicates that the function's growth is slowing down because of discarding a steeper slope.

Understanding how a function grows is crucial in predicting its behavior in different intervals. In this exercise, we're tasked with finding how much the function value could increase between \(x = 5\) and \(x = 7\).
Initially, considering the first derivative \(f'(5) = 2\), if the derivative were constant (which it is not), the function value may increase by 4 units over the interval. So we might expect \(f(7)\) to be \(24\), starting from \(f(5) = 20\). However, due to the concavity, the actual increase is less than 4.

Let's apply this to the options:

• For \(f(7) = 26\), this result would signify an increase of 6 units, which is impossible due to the decreasing rate of growth.
• If \(f(7) = 24\), this is the maximum possible increase, making it a feasible outcome.
• For \(f(7) = 22\), this reflects a lesser increase than the maximum possible, allowing it to be a possible solution.

In summary, analyzing how the concavity impacts function growth gives insight into how values may change over intervals and helps determine possible outcomes.