百度作业网设f'(x)∫(0,2)f(x)dx=50,且f(0)=0,f(x)≥0,求∫(0,2)f(x)dx及f(x)
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设f'(x)∫(0,2)f(x)dx=50,且f(0)=0,f(x)≥0,求∫(0,2)f(x)dx及f(x)
f'(x)是个函数
∫(0 ,2) f(x) dx是个数值
而f'(x) * ∫(0 ,2) f(x) dx又是个数值,则f'(x)必是常数函数
即设f(x) = ax + b ,f'(x) = a ,x ≥ 0
a * ∫(0 ,2) (ax + b) dx = 50
a * [ax^2/2 + bx](0 ,2) = 50
a * [a(2)^2/2 + b(2)] = 50
a * (2a + 2b) = 50
a^2 + ab = 25
又f(0) = 0
==> f(0) = a(0) + b = 0 ==> b = 0
a^2 = 25
a = |5|
f(x) ≥ 0 ==> f(x) = 5|x| ,x ≥ 0
∫(0 ,2) f(x) dx = 10
∫(0 ,2) f(x) dx是个数值
而f'(x) * ∫(0 ,2) f(x) dx又是个数值,则f'(x)必是常数函数
即设f(x) = ax + b ,f'(x) = a ,x ≥ 0
a * ∫(0 ,2) (ax + b) dx = 50
a * [ax^2/2 + bx](0 ,2) = 50
a * [a(2)^2/2 + b(2)] = 50
a * (2a + 2b) = 50
a^2 + ab = 25
又f(0) = 0
==> f(0) = a(0) + b = 0 ==> b = 0
a^2 = 25
a = |5|
f(x) ≥ 0 ==> f(x) = 5|x| ,x ≥ 0
∫(0 ,2) f(x) dx = 10
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