Problem 13 - Calculus for Biology and Medicine

Question

Show that if \(f(x)\) is a positive twice-differentiable function that has a local minimum at \(x=c\), then \(g(x)=[f(x)]^{2}\) has a local minimum at \(x=c\) as well.

Step-by-Step Solution

Answer

If \(f(x)\) has a local minimum at \(x = c\), then \(g(x) = [f(x)]^2\) also has a local minimum at \(x = c\).

Step 1: Understand the Problem

We are given that the function \(f(x)\) has a local minimum at \(x = c\). We need to prove that \(g(x) = [f(x)]^2\) also has a local minimum at \(x = c\). This involves using properties of derivatives since both functions are twice-differentiable.

Step 2: Analyze the Derivative of f(x)

Since \(f(x)\) has a local minimum at \(x = c\), the first derivative \(f'(c) = 0\), and the second derivative \(f''(c) \geq 0\) for positivity of the function.

Step 3: Differentiate g(x)

Compute the first derivative of \(g(x)\): \[ g'(x) = 2f(x)f'(x). \] This derivative will be used to examine \(g(x)\) around \(x=c\).

Step 4: Evaluate g'(x) at x = c

Substitute \(x = c\) into \(g'(x) = 2f(x)f'(x)\): \[ g'(c) = 2f(c)f'(c). \] Given \(f'(c) = 0\), it follows that \(g'(c) = 0\).

Step 5: Compute the Second Derivative of g(x)

The second derivative of \(g(x)\) is \[ g''(x) = 2[(f'(x))^2 + f(x)f''(x)]. \] This needs to be evaluated at \(x = c\) to determine the nature of the critical point.

Step 6: Evaluate g''(x) at x = c

Substitute \(x = c\) into \(g''(x)\): \[ g''(c) = 2[(f'(c))^2 + f(c)f''(c)]. \] Since \(f'(c) = 0\), it simplifies to \[ g''(c) = 2f(c)f''(c). \]

Step 7: Determine the Sign of g''(c)

We know \(f(x)\) is positive, so \(f(c) > 0\). Also, \(f''(c) \geq 0\) because \(f(x)\) has a local minimum at \(x = c\). Hence, \(g''(c) = 2f(c)f''(c) \geq 0\).

Step 8: Conclude the Nature of g(x) at x = c

Since \(g'(c) = 0\) and \(g''(c) \geq 0\), \(g(x)\) has a local minimum at \(x = c\). Thus, we've shown that \([f(x)]^2\) also possesses a local minimum there.

Key Concepts

Understanding Local Minimum

When we say that a function has a "local minimum" at a point, it means that around this point, the function takes on values that are not lower than its value at that point. Imagine standing in a small valley among hills; you're at a local minimum because you are lower than the nearby points.
For any function to have a local minimum, specifically at a point like \( x = c \), a couple of conditions must be satisfied:

• The first derivative of the function, \( f'(c) \), must be zero. This indicates a flat slope at that point, like the very bottom of a hill or valley.
• The second derivative of the function, \( f''(c) \), must be greater than or equal to zero. This second condition ensures the curve is concave upward at \( c \), reinforcing it as a minimum.

In our original exercise, since \( f(x) \) has a local minimum at \( x = c \), these conditions are already established. This sets the stage for determining if \( g(x) = [f(x)]^2 \) will also have a local minimum there.

The Role of Derivatives

Derivatives are powerful tools in calculus that help us understand how functions change. They tell us the slope of a function at any given point. The first derivative, \( f'(x) \), helps find critical points. These are points where the function could have a maximum, minimum, or some kind of "flat" point.
Because the local minimum for \( f(x) \) is at \( x=c \), we set \( f'(c)=0 \). This tells us the slope at \( x=c \) is flat, affirming \( c \) is a special point.
The second derivative, \( f''(x) \), provides information about the shape around the point. If \( f''(c) > 0 \), the graph opens upwards, and it's confirmed a local minimum. If \( f''(c) < 0 \), the graph opens downwards, and it’s potentially a local maximum. Thus, \( f''(c) \geq 0 \) ensures \( x=c \) is a local minimum.For the function \( g(x) = [f(x)]^2 \), analyzing the first derivative confirms critical points, while examining the second derivative helps ensure that a local minimum exists as well.

Exploring Twice-Differentiable Functions

A "twice-differentiable function" means that the function can have two derivatives that are both continuous. This is crucial when dealing with functions such as \( f(x) \) in calculus because:

• The continuity of the first derivative secures that the function is smooth, with no sharp corners or breaks.
• The second derivative's existence and continuity ensure we can study the concavity and points of inflection with accuracy.

In our exercise, the fact that \( f(x) \) is twice-differentiable supports the approach to discover its local behavior—specifically at \( x=c \).
This allows us to determine not just that \( g(x) = [f(x)]^2 \) also is twice differentiable, but to confidently find both \( g'(c) \) and \( g''(c) \) as we did, confirming the local minimum.
This continuous differentiability is essential for ensuring the function behaves predictably and the calculations for the derivatives are seamless and accurate.