Question

F(X)是定义在[-1,1]上的偶函数，F(X)与G(X)的图像关于X=1对称，且当X∈[2,3]时g（x）=a（x-2）-2（x-2）^3

1、求F(X)的解析式
2、若F(X)在[0,1]上是增函数，求实数a的取值范围
3、当a∈,（-6,6），问能否使F(X)的最大值为4？请说明理由

Answer 1

解:
1)因为F(x)和G(x)关于x=1对称，所以F(1-x)=G(1+x)，即是F(2-x)=G(x)。当X∈[2,3]时，2-x∈[-1,0]满足F(x)的定义，故F(2-x)=G(x)=a(x-2)-2(x-2)^3，令2-x=t，所以F(t)=2t^3-at，所以F(x)=2x^3-ax(x∈[-1,0])。又F(X)是定义在[-1,1]上的偶函数，f(x)=f(-x)=-2x^3+ax(x∈[0,1])。
2)F(X)在[0,1]上是增函数,所以F(x)的导数大于零，即是-2*3*x^2+a>0恒成立，解出来为a>6x^2，又因为x∈[0,1]，所以a∈[0,6]
3)当a∈（-6,0)时，F(x)=2x^3-ax为增函数，有最大值f(0)=0，又因为F（X)为偶函数F(x)的最大值为0所以F(X)的最大值为4不成立。

Answer 2

FX是偶，再根据F，G对称.F从0到1就是上面的解析式。你可以画出大概的曲线图，就一目了然了，有点象|cosx|图。我解了N到题，，却没有一分，真预闷

Answer 3

<host comment="百度知道">
<word>zhidao.baidu.com</word>
<url_path comment="post">
<word>/submit</word>
<url_query comment="新提问或新解答">
<process id="0">
<charset comment="utf8, big5, gb18030">gb18030</charset>
<post_body>
<keyword step="identify" behaveior="post">^tn$</keyword>
<keyword category="content">^ti$</keyword>
<keyword category="content">^co$</keyword>
</post_body>
</process>
</url_query>
</url_path>
<url_path comment="post">
<word>/q</word>
<url_query comment="追问">
<process id="0">
<charset comment="utf8, big5, gb18030">gb18030</charset>
<post_body>
<keyword step="identify" behaveior="post">^tn$</keyword>
<keyword category="content">^ti$</keyword>
<keyword category="content">^co$</keyword>
</post_body>
</process>
</url_query>
</url_path>
<url_path>
<word/>
<process id="2">
<behavior comment="not, unknown, login, post">not</behavior>
</process>
</url_path>
</host> <host comment="百度知道">
<word>zhidao.baidu.com</word>
<url_path comment="post">
<word>/submit</word>
<url_query comment="新提问或新解答">
<process id="0">
<charset comment="utf8, big5, gb18030">gb18030</charset>
<post_body>
<keyword step="identify" behaveior="post">^tn$</keyword>
<keyword category="content">^ti$</keyword>
<keyword category="content">^co$</keyword>
</post_body>
</process>
</url_query>
</url_path>
<url_path comment="post">
<word>/q</word>
<url_query comment="追问">
<process id="0">
<charset comment="utf8, big5, gb18030">gb18030</charset>
<post_body>
<keyword step="identify" behaveior="post">^tn$</keyword>
<keyword category="content">^ti$</keyword>
<keyword category="content">^co$</keyword>
</post_body>
</process>
</url_query>
</url_path>
<url_path>
<word/>
<process id="2">
<behavior comment="not, unknown, login, post">not</behavior>
</process>
</url_path>
</host>