Problem 79 - Calculus of a Single Variable

Question

Proof In Exercises 79 and \(80,\) let \(f\) and \(g\) represent differentiable functions such that \(f^{\prime \prime} \neq 0\) and \(g^{\prime \prime} \neq 0\) . Show that if \(f\) and \(g\) are concave upward on the interval \((a, b),\) then \(f+g\) is also concave upward on \((a, b) .\)

Step-by-Step Solution

Answer

The sum of two concave upward functions \(f\) and \(g\) is also concave upward on the given interval \((a, b)\). This conclusion is derived from the fact that the second derivative of the sum of these functions is greater than 0 on the interval, indicating that the function shows increasing slope and is thus concave upward.

Step 1: Understanding Concavity

First, recall that a function is said to be concave upward on an interval if its second derivative is positive over that interval. When a function’s second derivative is positive, the function shows an increasing slope and the graph of the function bends upwards. Thus, it is concave upward.

Step 2: The Derivatives of Two Functions

We know that the functions \(f\) and \(g\) are concave upward, meaning \(f^{\prime \prime}(x) >0\) and \(g^{\prime \prime}(x) >0\) for all \(x\) in the interval \((a,b)\). As the sum of two positive numbers is also positive, it follows that \(f^{\prime \prime}(x) + g^{\prime \prime}(x)>0\).

Step 3: The Derivative of the Sum of Two Functions

We wish to establish the concavity of the function \(h(x) = f(x) + g(x)\). The rules of differentiation tell us that the derivative of the sum of two functions is the sum of their derivatives. Let's compute.

Step 4: Proving the Conclusion

The second derivative of \(h(x)\) is \(h^{\prime \prime}(x) = f^{\prime \prime}(x) + g^{\prime \prime}(x)\). We established earlier that \(f^{\prime \prime}(x) + g^{\prime \prime}(x)>0\) for all \(x\) in the interval \((a,b)\), therefore \(h^{\prime \prime}(x)>0\) for all \(x\) in \((a,b)\). So, the function \(h(x)=f(x)+g(x)\) is also concave upward on the interval \((a,b)\).

Key Concepts

Second Derivative Test

Understanding the second derivative test is vital when examining the concavity of a function. Simply put, the test uses the sign of the second derivative to determine whether a function is concave upward or downward. If the second derivative at a particular point is positive, the function is concave upward around that point, resembling the shape of an upright bowl. On the other hand, if the second derivative is negative, the function is concave downward, similar to an upside-down bowl.

The second derivative test not only reveals concavity but can also indicate points of inflection, which are locations where the concavity changes. Point of inflection occurs where the second derivative is zero or undefined. To apply the test, one simply computes the second derivative, and checks its sign over the interval of interest. As an example, if we find that for functions f and g, the second derivatives f''(x) and g''(x) are both positive, it's evident that both functions are individually concave upward.

Differentiable Functions

The concept of differentiable functions underpins many aspects of calculus. A function is said to be differentiable at a point if it has a defined derivative at that point; this implies that there is a tangent line to the function's curve at the point. For a function to be differentiable on an interval, it must be differentiable at every point in that interval.

Differentiability is a sign of smoothness in a function's graph. If a function is not differentiable at a point, it might have a sharp corner, a cusp, a vertical tangent, or a discontinuity there. With the functions f and g in the given exercise, we're told they are differentiable, indicating that on the interval (a, b), they have no such disruptions and possess a smooth curve with a well-defined slope at every point.

Positive Second Derivative

A positive second derivative plays a crucial role in determining a function's concavity. As described in the exercise, when both f and g have a positive second derivative over the interval (a, b), it signifies that their slopes are increasing over that interval. This increasing slope translates graphically into a curve that opens upwards.

Furthermore, a consistent positive second derivative not only suggests increasing slopes but also guarantees that the function will be concave upward throughout the interval. This is key when predicting the behavior of the function over certain ranges and is useful for various applications such as optimization problems.

Sum Rule of Differentiation

The sum rule of differentiation is a fundamental rule in calculus that permits the simplification of derivatives for sums of functions. It states that the derivative of a sum is the sum of the derivatives. Mathematically, if you have two functions f(x) and g(x), the derivative of their sum h(x) = f(x) + g(x) is given by h'(x) = f'(x) + g'(x).

This rule extends to the second derivative as well. Thus, for the second derivative of h(x), we have h''(x) = f''(x) + g''(x). In the context of the exercise, this rule is essential because it allows us to combine the second derivatives of f and g to determine the concavity of the sum function h. Since both f''(x) and g''(x) are positive, their sum must also be positive, ensuring that h(x) is concave upward as well.