Cho hàm số $f\left( x \right)={{\log }_{2}}\left(...

The Knowledge · 23/2/22

Câu hỏi: Cho hàm số

f (x) = \log_{2} (x - \frac{1}{2} + \sqrt{x^{2} - x + \frac{17}{4}})

. Tính

T = f (\frac{1}{2021}) + f (\frac{2}{2021}) + . . . + f (\frac{2020}{2021})

A.

T = 2021

.
B.

T = 2019

.
C.

T = 2018

.
D.

T = 2020

.

Ta có:

f (1 - x) = \log_{2} (1 - x - \frac{1}{2} + \sqrt{{(1 - x)}^{2} - (1 - x) + \frac{17}{4}}) = \log_{2} (\sqrt{x^{2} - x + \frac{17}{4}} - (x - \frac{1}{2}))

f (x) + f (1 - x) = \log_{2} (x - \frac{1}{2} + \sqrt{x^{2} - x + \frac{17}{4}}) + \log_{2} (\sqrt{x^{2} - x + \frac{17}{4}} - (x - \frac{1}{2}))

= \log_{2} [(x - \frac{1}{2} + \sqrt{x^{2} - x + \frac{17}{4}}) (\sqrt{x^{2} - x + \frac{17}{4}} - (x - \frac{1}{2}))]

= \log_{2} 4 = 2

\Rightarrow T = f (\frac{1}{2021}) + f (\frac{2}{2021}) + . . . + f (\frac{2020}{2021})

= f (\frac{1}{2021}) + f (\frac{2020}{2021}) + f (\frac{2}{2021}) + f (\frac{2019}{2021}) + . . . + f (\frac{1010}{2021}) + f (\frac{1011}{2021})

= 1010.2 = 2020

.

Đáp án D.