Example Theorem1.Jf0≤x≤2,then-x3+4x+1>0. Before we write a proof of this theorem, we have to do some scratchwork to figure out why it is true The inequality certainly holds for a =0; then the left side is equal to 1 and 1>0.As grows, the 4r term(which is positive) initially seems to have greater magnitude than (which is negative). For example, when a= l, we have 4r=4, but- 3=-1 only. In fact, it looks like doesn t begin to dominate until r>2. So it seems the -+4c part should be nonnegative for all z between 0 and 2, which would imply that -r+4.r+1 So far, so good. But we still have to replace all those"seems like"phrases with solid, logical arguments. We can get a better handle on the critical- r +4.r part by factoring it, which is not too hard 4x2=x(2-x)(2+x) Aha! For r between 0 and 2, all of the terms on the right side are nonnegative. And a product of nonnegative terms is also nonnegative. Let's organize this blizzard of obser vations into a clean proof Proof. Assume0 <a <2. Then a, 2-, and 2+3 are all nonnegative. Therefore, the Cumber so. these terms is also nonnegative. Adding, to this product gives a positive r(2-x)(2+x)+1>0 Multiplying out on the left side proves that x3+4x+1>0 as claimed There are a couple points here that apply to all proofs You'll often need to do some scratchwork while you're trying to figure out the lsso y ical steps of aproof. Your scratchwork can be as disorganized as you like-full dead-ends, strange diagrams, obscene words, whatever. But keep your scratchwork separate from your final proof, which should be clear and concise Proofs typically begin with the word"Proof"and end with some sort of doohickey like d or"qe d". The only purpose for these conventions is to clarify where proof begin and end6 Proofs Example Theorem 1. If 0 ≤ x ≤ 2, then −x3 + 4x + 1 > 0. Before we write a proof of this theorem, we have to do some scratchwork to figure out why it is true. The inequality certainly holds for x = 0; then the left side is equal to 1 and 1 > 0. As x 3 grows, the 4x term (which is positive) initially seems to have greater magnitude than −x 3 (which is negative). For example, when x = 1, we have 4x = 4, but −x = −1 only. In fact, it looks like −x3 doesn’t begin to dominate until x > 2. So it seems the −x3 + 4x part should be nonnegative for all x between 0 and 2, which would imply that −x3 + 4x + 1 is positive. So far, so good. But we still have to replace all those “seems like” phrases with solid, logical arguments. We can get a better handle on the critical −x3 + 4x part by factoring it, which is not too hard: 2 −x 3 + 4x = x(2 − x)(2 + x) Aha! For x between 0 and 2, all of the terms on the right side are nonnegative. And a product of nonnegative terms is also nonnegative. Let’s organize this blizzard of obser￾vations into a clean proof. Proof. Assume 0 ≤ x ≤ 2. Then x, 2 − x, and 2 + x are all nonnegative. Therefore, the product of these terms is also nonnegative. Adding 1 to this product gives a positive number, so: x(2 − x)(2 + x) + 1 > 0 Multiplying out on the left side proves that −x 3 + 4x + 1 > 0 as claimed. There are a couple points here that apply to all proofs: • You’ll often need to do some scratchwork while you’re trying to figure out the log￾ical steps of aproof. Your scratchwork can be as disorganized as you like— full of dead­ends, strange diagrams, obscene words, whatever. But keep your scratchwork separate from your final proof, which should be clear and concise. • Proofs typically begin with the word “Proof” and end with some sort of doohickey like � or “q.e.d”. The only purpose for these conventions is to clarify where proofs begin and end