Proof of The Second Derivative Test
Suppose we have (the proof for is analogous). Then
Thus, for h sufficiently small we get
which means that if h < 0 so that f is decreasing to the left of x, and that if h > 0 so that f is increasing to the right of x.
Now, by the first derivative test we know that has a local minimum at .
Read more about this topic: Second Derivative Test
Famous quotes containing the words proof of the, proof of, proof, derivative and/or test:
“If we view our children as stupid, naughty, disturbed, or guilty of their misdeeds, they will learn to behold themselves as foolish, faulty, or shameful specimens of humanity. They will regard us as judges from whom they wish to hide, and they will interpret everything we say as further proof of their unworthiness. If we view them as innocent, or at least merely ignorant, they will gain understanding from their experiences, and they will continue to regard us as wise partners.”
—Polly Berrien Berends (20th century)
“A short letter to a distant friend is, in my opinion, an insult like that of a slight bow or cursory salutation—a proof of unwillingness to do much, even where there is a necessity of doing something.”
—Samuel Johnson (1709–1784)
“The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
—Andrew Michael Ramsay (1686–1743)
“Poor John Field!—I trust he does not read this, unless he will improve by it,—thinking to live by some derivative old-country mode in this primitive new country.... With his horizon all his own, yet he a poor man, born to be poor, with his inherited Irish poverty or poor life, his Adam’s grandmother and boggy ways, not to rise in this world, he nor his posterity, till their wading webbed bog-trotting feet get talaria to their heels.”
—Henry David Thoreau (1817–1862)
“A test of what is real is that it is hard and rough. Joys are found in it, not pleasure. What is pleasant belongs to dreams.”
—Simone Weil (1909–1943)