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# 12.08 - [Hand out cover sheet for exam and read it aloud...

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[Hand out cover sheet for exam and read it aloud] The exam will focus on sections 2.6 to 4.7. You may use a five-page “cheat-sheet”, but it must be created by you; you cannot photocopy reference books or print things out from the web. I aim to arrive here shortly before 11; as soon as everybody’s here, we’ll start the exam. You can stay till 1 p.m. Return HW #10: “There exists a function f such that f ( x ) < 0, f ( x ) < 0, and f ′′ ( x ) > 0 for all x .” ..?.. #10½ (Extra True/False question): Does there exist a twice- differentiable function f on R satisfying f ( x ) < 0 and f ′′ ( x ) > 0 for all x ? ..?.. Use the original definition of concavity. ..?.. Since f ′′ ( x ) > 0 for all x , the function f must be concave up everywhere, so every tangent line to the graph of f must

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lie completely below the graph of f . And since the graph of f stays below the x -axis, every tangent line to the graph of f must lie completely below the x -axis. Hence … ..?..
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