Question

设f(x)=e^ax(a>0).过点P(a,0)且平行于y轴的直线与曲线C:y=f(x)的交点为Q,曲线C过点Q的切线交

Answer 1

B

f(x) = e^(ax)

Q(a,e^(a²))

f'(x) = ae^(ax)

f'(a) = ae^(a²)

过点Q的切线:y - e^(a²) = ae^(a²)(x - a)

y = 0,x = a - 1/a,R(a - 1/a,0)

三角形PQR的面积S = (1/2)RP*PQ

= (1/2)(a - a + 1/a)*e^(a²)

= [e^(a²)]/(2a)

对a求导:S' = 2[e^(a²)](2a² - 1)/a² = 0

a² = 1/2

a = 1/√2

S = √(2e)/2