Re: [微分] 兩題證明
第二題照反過來做第一題就好囉！
The def of concave up:all the points in an open interval I should be
above any tangent lines in this interval.
In other words,f'(x) should be increasing in this interval.
Consider an open interval I:(a,b) b>a and f(x) is define on any x belongs to I
Set a tangent line y=f(a)+f'(a)(x-a)
Then by Mean value theorem there is a number c between (a,b)
such that f(x)-f(a)=f'(c)(x-a)
since f''(x)>0 for any x belongs to (a,b) f'(x) is increasing on I.
f'(a)<f'(c) then multipling the both sides by x-a (>0)
f'(a)(x-a)<f'(c)(x-a) then adding bothsides by f(a)
f(a)+f'(a)(x-a)<f(a)+f'(c)(x-a)
so we got y>f(a)+f'(a)(x-a)
Q.E.D.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 122.127.33.35
討論串 (同標題文章)
完整討論串 (本文為第 2 之 2 篇)：