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)
“The fact that several men were able to become infatuated with that latrine is truly the proof of the decline of the men of this century.”
—Charles Baudelaire (1821–1867)
“It comes to pass oft that a terrible oath, with a swaggering accent sharply twanged off, gives manhood more approbation than ever proof itself would have earned him.”
—William Shakespeare (1564–1616)
“When we say “science” we can either mean any manipulation of the inventive and organizing power of the human intellect: or we can mean such an extremely different thing as the religion of science the vulgarized derivative from this pure activity manipulated by a sort of priestcraft into a great religious and political weapon.”
—Wyndham Lewis (1882–1957)
“The test of a real comedian is whether you laugh at him before he opens his mouth.”
—George Jean Nathan (1882–1958)