中4代數函數題目

條題目係話：
f(x) = y = a^x, 而x最細值係0，最大值係1。
而家，y最細值+y最大值=3，咁a係咩？

ANSWER:
Since f(x) is a monotonic function (only incrasing or decreasing) for x E[0,1],
so, put x(min) and x(max) into y,
y will become min and max respectively, (for f(x) is increasing), or
y will become max and min respectively, (for f(x) is decreasing).

Thus,
put x=0
y=a^0=1 ---(1)
put x=1
y=a^1=a ---(2)
(1)+(2): 1+a=3, so a=2.

2008-06-06 09:42:27 補充：
Also, it is a monotonic increasing function:
x increases, y increases.

If a<1, f(x) is monotonic decreasing function.

2008-06-06 09:51:59 補充：
至於樓上位朋友，
prove f(x)係咪increasing嘅時候就似乎同我有出入喇，因為佢冇consider到如果a係純小數（即0至1之間嘅小數）。

y=a^x
lny=xlna
(dy/dx)(1/y)=lna
dy/dx=ylna=a^x (lna)&gt;0

so, it is increasing from 0 to 1
then, x=1 is largest, x=0 is smallest
when x=0 , y=1, (smallest)
when x=1, y=a, (largest,)

最大值和最少值之和
a 1=3
a=2