Determine whether the statements are true or false. If a...

Determine whether the statements are true or false. If a statement is false, find functions for which the statement fails to hold.
(a) If $f$ and $g$ are concave up on an interval, then so is $f+g .$
(b) If $f$ and $g$ are concave up on an interval, then so is $f \cdot g$

Key Concepts

Concavity

Concavity is a property of a function that describes the curvature of its graph. A function is said to be concave up on an interval if its second derivative is nonnegative throughout that interval. This means that the slope of the function is increasing and that the graph of the function lies below any of its tangent lines, demonstrating an upward bending shape.

Addition of Functions and Preservation of Concavity

When two functions are added together, their derivatives add as well. For twice-differentiable functions, the second derivative of the sum is the sum of the individual second derivatives. Therefore, if both functions are concave up (with nonnegative second derivatives) on an interval, their sum will also have a nonnegative second derivative, guaranteeing that the sum is concave up.

Multiplication of Functions and Concavity

Multiplying functions is more complex than addition because the product rule for derivatives introduces additional terms that mix the functions and their derivatives. These extra terms can affect the sign of the second derivative, meaning that the product of two concave up functions is not necessarily concave up. This concept highlights that the property of concavity does not always carry over under multiplication, and counterexamples can be constructed to show instances where the product fails to be concave up.

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