若f(x)在实数域内二阶可导,f(x)=-f(-x)且在0到正无穷内有f'(x)>0,f''(x)>0,则f(x)在负无穷到0内f'(x)f"(x)如何?

Question

若f(x)在实数域内二阶可导,f(x)=-f(-x)且在0到正无穷内有f'(x)>0,f''(x)>0 ,则f(x)在负无穷到0内f'(x)f"(x)如何?

Answer 1

由f(x)=-f(-x)得,f'(x)=[f(x)]'=[-f(-x)]'=-[f(-x)]'=-[-f'(-x)]=f'(-x),
即f'(x)=f'(-x).
从而,f"(x)=[f'(x)]'=[f'(-x)]'=-f''(-x)
当x0,由已知条件可知f'(-x)>0,f''(-x)>0,
于是,f'(x)=f'(-x)>0;
f"(x)=-f''(-x)