Problem 25 - Calculus for Biology and Medicine

Question

Suppose that \(f\) is twice differentiable on an open interval \(I\). Show that if \(f^{\prime \prime}(x)<0\), then \(f\) is concave down.

Step-by-Step Solution

Answer

If \(f''(x) < 0\), then \(f\) is concave down on interval \(I\) because the slope decreases.

Step 1: Define Concave Down

A function \(f\) is concave down on an interval \(I\) if for any two points \(x_1\) and \(x_2\) in \(I\), the function \(f\) lies below the line segment connecting \(f(x_1)\) and \(f(x_2)\). This means that the slope of the tangent line at any point is decreasing.

Step 2: Use Second Derivative Test

The second derivative \(f''(x)\) tells us about the concavity of the function. Specifically, if \(f''(x) < 0\) for all \(x\) in interval \(I\), it indicates that the rate of change of the slopes of the tangent lines (given by \(f'(x)\)) is negative.

Step 3: Interpret the Second Derivative

Since \(f''(x) < 0\), this means that the first derivative \(f'(x)\) is decreasing over the interval \(I\). As you move from left to right on the graph, the slope gets smaller, so the graph curves downward.

Step 4: Conclude Concavity

Because the second derivative is negative throughout the interval, and the first derivative \(f'(x)\) is decreasing, it follows directly that the function \(f\) is concave down on interval \(I\). This is consistent with the definition of concavity.

Key Concepts

Second Derivative Test

The second derivative test is a valuable tool in calculus to determine the concavity of a function. When you have a function that is twice differentiable, you can find its second derivative, denoted as \(f''(x)\). This test uses the sign of the second derivative to inform us about the function's behavior.
If \(f''(x) > 0 \) on an interval, the function \( f \) is concave up on that interval, meaning its graph is curving upwards like a cup. Conversely, if \(f''(x) < 0 \), the function is concave down, and the graph curves down like an upside-down cup.

• The second derivative tells us about how the rate of change of the first derivative \(f'(x)\) behaves.
• In simpler terms, if the second derivative is positive, the slope of the function (or the rate of increase) is getting steeper. If negative, it's "sliding" downwards, meaning the slope of the function is getting smaller or less steep.

This test provides a straightforward way to identify the nature of a graph without plotting every point.

Twice Differentiable Functions

When we say a function is twice differentiable, we mean it has both a first and a second derivative. In calculus, a differentiable function like this is smooth without any sharp turns or corners. The ability to take a second derivative signals that the function is sufficiently smooth for in-depth analysis.

• A twice differentiable function \( f \) on an interval \( I \) means both the function \( f \) and its first derivative \( f'(x) \) are expressions that can be differentiated once more.
• For example, \( f(x) = x^3 \) is twice differentiable because both its first and second derivatives, \( f'(x) = 3x^2 \) and \( f''(x) = 6x \), exist for all real numbers.

This property is essential when applying tests like the second derivative test because it guarantees continuity and smoothness of the function and its slopes.

Concave Down Definition

A function is described as concave down on a particular interval if it arcs downwards. Imagine a parachute’s shape; that's how a concave down graph appears. This occurs when for every two distinct points on the interval, the function's graph is below the straight line joining those points. This creates a "bowl-like" appearance that flips upside down.

• The mathematical condition \( f''(x) < 0 \) provides a criterion for a function to be concave down.
• A concave down graph means any tangent line will be above the curve itself, and as you move along the curve, these tangents' slopes become less steep.
• In algebraic terms, as \( x \) increases, \( f'(x) \) decreases.

Understanding concave down curves is vital in analyzing graphs, assisting in predictions, and gaining insights about behavior over intervals.