Problem 44 - Calculus for Biology and Medicine

Question

Assume that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b) .\) Show that if \(f(a)

Step-by-Step Solution

Answer

By the Mean Value Theorem, there is a point between \(a\) and \(b\) where the derivative \(f'\) is positive.

Step 1: Understand the Conditions

The problem gives a continuous function \( f \) on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\). It also states that \( f(a) < f(b) \). We need to investigate the behavior of the derivative \( f' \) in \((a, b)\).

Step 2: Recall the Mean Value Theorem

The Mean Value Theorem (MVT) states that for a function \( f \) that is continuous on \([a, b]\) and differentiable on \((a, b)\), there exists at least one point \( c \) in \((a, b)\) such that \( f'(c) = \frac{f(b)-f(a)}{b-a} \).

Step 3: Apply the Mean Value Theorem

Since \( f(a) < f(b) \), the difference \( f(b) - f(a) \) is positive. The MVT guarantees that there is a point \( c \) in \((a, b)\) such that \( f'(c) = \frac{f(b) - f(a)}{b-a} \). Since \( f(b) - f(a) > 0 \) and \( b-a > 0 \), it follows that \( f'(c) > 0 \).

Step 4: Conclusion

The existence of this point \( c \) where \( f'(c) > 0 \) confirms that the function is increasing at some point between \( a \) and \( b \). Therefore, we have shown that \( f' \) is positive at some point in \((a, b)\).

Key Concepts

Continuous Function

To understand continuous functions, imagine tracing a path on a graph without lifting your pen. This path, or function, is continuous when it can be drawn without any jumps, gaps, or sudden changes. Mathematically, a function \( f \) is continuous on an interval \([a, b]\) if it has no interruptions in that interval.

• The graph of a continuous function has no breaks within \([a, b]\).
• The function values approach the same value from either side of any point within the interval.
• Continuity at a point \( x \) means \( \lim_{{x \to c}} f(x) = f(c) \).

These properties of continuous functions on \([a, b]\) allow us to use powerful mathematical tools like the Mean Value Theorem. Without continuity, such theorems and predictions would fail, as they rely on smooth transitions between values.

Differentiable Function

Differentiability of a function puts a stricter condition than continuity. If a function is differentiable, it is also continuous, but not vice-versa. Differentiability on an interval \((a, b)\) means that the function has a defined tangent at each point in this interval.

• A differentiable function has a defined derivative \( f'(x) \) at each point in its domain.
• In practical terms, the function's graph has no sharp corners or cusps within this interval.
• It also means small changes in input near any point produce small changes in output without any sudden twists.

So, for a function to be differentiable, it must be smooth and have predictable rates of change. In the context of the Mean Value Theorem, differentiability ensures that we can calculate the exact rate of change at least at some point \( c \) within \((a, b)\).

Derivative

The derivative of a function reflects its rate of change. Think of it as the function's velocity, showing how quickly and in what direction the function value is changing at any given point. If you know the derivative \( f'(x) \) of a function \( f(x) \), you can understand how \( f(x) \) behaves at small scales.

• The derivative \( f'(x) \) measures the slope of the tangent line to the curve at any point \( x \).
• A positive derivative indicates an increasing function, like climbing a hill.
• A negative derivative indicates a decreasing function, akin to going downhill.
• When the derivative is zero, the function is flat at that point, suggesting a peak, trough, or a constant segment.

In our exercise, the Mean Value Theorem ensures that if a continuous and differentiable function starts lower at \( a \) than it is at \( b \), the derivative must be positive at some point between \( a \) and \( b \), indicating an upward trend.