Problem 44 - Calculus with Concepts in Calculus

Question

Show by giving an example that the graph of the function \(f g\) need not be concave upward on an open interval \(I\) even if the graph of \(f\) is concave upward on \(I\) and the graph of \(g\) is concave upward on \(I .\)

Step-by-Step Solution

Answer

The function \(x^4\) is not concave upward at \(x = 0\).

Step 1: Define a Concave Upward Function

A function is concave upward on an interval if its second derivative is non-negative on that interval. Let's consider the function \(f(x) = x^2\), which is concave upward since \(f''(x) = 2 > 0\) for all \(x\).

Step 2: Define Another Concave Upward Function

Now, define another function \(g(x) = x^2\). This function is also concave upward on any interval because \(g''(x) = 2 > 0\) for all \(x\).

Step 3: Calculate the Product Function

Evaluate the product of the two functions: \((f \cdot g)(x) = f(x) \cdot g(x) = x^2 \cdot x^2 = x^4\).

Step 4: Determine Concavity of the Product Function

Calculate the second derivative of the product function. Take the first derivative: \((f \cdot g)'(x) = 4x^3\). Then, the second derivative is \((f \cdot g)''(x) = 12x^2\), which is zero at \(x = 0\). Hence, the graph of \(f \cdot g\) (i.e., \(x^4\)) is not concave upward at \(x = 0\).

Step 5: Conclusion

Even though both \(f(x) = x^2\) and \(g(x) = x^2\) are concave upward individually, the product function \(x^4\) is not concave upward at least at \(x=0\). This provides an example where the product of two concave upward functions is not concave upward on an interval.

Key Concepts

Second Derivative

The second derivative is a powerful tool that helps us understand the shape and behavior of a graph. To determine whether a function is concave upward or concave downward, examining the second derivative is essential.
It is obtained by differentiating the first derivative.

• If the second derivative of a function is positive over an interval, it indicates that the graph is concave upward in that region.
• If it's negative, the graph is concave downward.
• If the second derivative is exactly zero at a point, the function might have a point of inflection there, which means the concavity could change.

Understanding and identifying these features help us grasp how the function behaves and predict its geometry.

Concave Upward Function

A concave upward function resembles a smiley face; it curves upward like a bowl. This type of graph bends upwards on the intervals where its second derivative is non-negative.
Concave upward functions include parabolas that open upwards like a U-shape.For instance:

• The function \( f(x) = x^2 \) is a perfect example of a concave upward function.
• Its second derivative, \( f''(x) = 2 \), is always positive, confirming its concavity.

Concave upward functions can help in various applications, including optimization problems that require finding local minima.

Product of Functions

When multiplying two functions together, it's called a product of functions. The result can have unpredictable concavity. Each function contributes to the final shape of the product.
Even if two functions are concave upward, multiplying them doesn't guarantee that their product will also be concave upward. In the exercise:

• We considered \( f(x) = x^2 \) and \( g(x) = x^2 \), both concave upward.
• However, their product \( (f \cdot g)(x) = x^4 \) has a second derivative \( 12x^2 \), and at \( x=0 \), it equals zero.

Thus, the product has regions where it is not concave upward, demonstrating the complexity involved when dealing with products of functions.