Is the statement true or false? You are told that f^'' and...

Is the statement true or false? You are told that $f^{\prime \prime}$ and $g^{\prime \prime}$ exist and that $f$ and $g$ are concave up for all $x .$ If a statement is true, explain how you know. If a statement is false, give a counterexample. $f(x)-g(x)$ cannot be concave up for all $x$.

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Key Concepts

Counterexample in Mathematical Proof

A counterexample is used to demonstrate that a general statement is false by presenting a specific case where the statement does not hold. In this context, presenting a case where two concave up functions yield a difference that is concave up disproves any assertion that the difference cannot be concave up.

Concavity

Concavity refers to the way a function curves. A function is said to be concave up if its second derivative is nonnegative at every point in its domain. This indicates that the graph of the function is ‘bent’ upward, and any line segment connecting two points on the graph lies above or on the graph.

Second Derivative Test

The second derivative test is used to assess the concavity of a function. If the second derivative of a function is positive for all values of x, the function is concave up. This test is a standard tool in calculus for understanding the curvature and shape of function graphs.

Operations on Functions (Subtraction)

When subtracting two functions, the second derivative of the resulting function is the difference of the second derivatives of the original functions. Even if both original functions are concave up, their difference can have varying concavity depending on the relative sizes of their second derivatives.

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