Cho hàm số y = f ( x ) có bảng biến thiên a) 2 ∫ 1 f ′ ( x ) d x = 1 .
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Cho hàm số \(y = f\left( x \right)\) có bảng biến thiên
a) \(\int\limits_1^2 {f'\left( x \right)dx} = 1\).
b) \(\int\limits_1^4 {\left[ {3 + f'\left( x \right)} \right]dx} = f\left( 4 \right) + 3\).
c) \(\int\limits_1^2 {\left| {f'\left( x \right)} \right|dx} = f\left( 1 \right) - f\left( 2 \right)\).
d) Nếu \(\int\limits_1^4 {\left| {f'\left( x \right)} \right|dx} = 5\) thì \(f\left( 4 \right) = 3\).
a) \(\int\limits_1^2 {f'\left( x \right)dx} = \left. {f\left( x \right)} \right|_1^2 = f\left( 2 \right) - f\left( 1 \right) = - 1 - 0 = - 1\).
b) \(\int\limits_1^4 {\left[ {3 + f'\left( x \right)} \right]dx} = \left. {3x} \right|_1^4 + \left. {f\left( x \right)} \right|_1^4 = 12 - 3 + f\left( 4 \right) - f\left( 1 \right) = 9 + f\left( 4 \right)\).
c) \(\int\limits_1^2 {\left| {f'\left( x \right)} \right|dx} = - \int\limits_1^2 {f'\left( x \right)dx} = \left. { - f\left( x \right)} \right|_1^2 = - f\left( 2 \right) + f\left( 1 \right)\).
d) \(\int\limits_1^4 {\left| {f'\left( x \right)} \right|dx} = 5\)\( \Leftrightarrow \int\limits_1^2 {\left| {f'\left( x \right)} \right|dx} + \int\limits_2^4 {\left| {f'\left( x \right)} \right|dx} = 5\)\[ \Leftrightarrow - f\left( 2 \right) + f\left( 1 \right) + \left. {f\left( x \right)} \right|_2^4 = 5\]\[ \Leftrightarrow - f\left( 2 \right) + f\left( 1 \right) + f\left( 4 \right) - f\left( 2 \right) = 5\]\[ \Leftrightarrow 1 + 0 + f\left( 4 \right) + 1 = 5 \Leftrightarrow f\left( 4 \right) = 3\].
Đáp án: a) Sai; b) Sai; c) Đúng; d) Đúng.