已知函数f(x)=a^x+x^2-xlna(a>0,a不等于1)⑴求函数f(x)在点(0,f(0))处的切线方程⑵求函数f(x)单调增区间
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![已知函数f(x)=a^x+x^2-xlna(a>0,a不等于1)⑴求函数f(x)在点(0,f(0))处的切线方程⑵求函数f(x)单调增区间](/uploads/image/z/12562074-18-4.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%3Da%5Ex%2Bx%5E2-xlna%EF%BC%88a%3E0%2Ca%E4%B8%8D%E7%AD%89%E4%BA%8E1%EF%BC%89%E2%91%B4%E6%B1%82%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%E5%9C%A8%E7%82%B9%EF%BC%880%2Cf%EF%BC%880%EF%BC%89%EF%BC%89%E5%A4%84%E7%9A%84%E5%88%87%E7%BA%BF%E6%96%B9%E7%A8%8B%E2%91%B5%E6%B1%82%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%E5%8D%95%E8%B0%83%E5%A2%9E%E5%8C%BA%E9%97%B4)
已知函数f(x)=a^x+x^2-xlna(a>0,a不等于1)⑴求函数f(x)在点(0,f(0))处的切线方程⑵求函数f(x)单调增区间
已知函数f(x)=a^x+x^2-xlna(a>0,a不等于1)
⑴求函数f(x)在点(0,f(0))处的切线方程
⑵求函数f(x)单调增区间
已知函数f(x)=a^x+x^2-xlna(a>0,a不等于1)⑴求函数f(x)在点(0,f(0))处的切线方程⑵求函数f(x)单调增区间
已知函数f(x)=a^x+x^2-xlna(a>0,a不等于1)
⑴求函数f(x)在点(0,f(0))处的切线方程
⑵求函数f(x)单调增区间
(1)解析:∵函数f(x)=a^x+x^2-xlna(a>0,a不等于1)
f(0)=1
函数f'(x)=a^x*lna+2x-lna==>f'(0)=0
∴函数f(x)在点(0,f(0))处的切线方程为y=1
(2)解析:f'(x)=a^x*lna+2x-lna=0==>x=0
f''(x)=a^x*(lna)^2+2>0
所以函数f(x)在x=0处取极小值
x<0时,函数f(x)单调减;
x>=0时,函数f(x)单调增;
(1)f'(x)=a^xIna 2x-Ina,f'(0)=Ina-Ina=0,故该切线的斜率为0,方程为y=f(0)=1
(2)f'(x)=0时,x=0,故f(x)在(0,正无穷)单调递增,在(负无穷,0)单调递减
1) f'(x)=a^x lna+2x-lna
f'(0)=lna+0-lna=0
f(0)=1
切线为y=1
2)f"(x)=a^x(lna)^2+2>0
因此f'(x)单调增,而f'(0)=0,
故当x>0时,f'(x)>0
所以f(x)的单调增区间为x>0