Question

Tìm GTLN - GTNN

B = \(\dfrac{4x+3}{x^2+1}\)

Answer 1

+) GTNN

B = \(\dfrac{4x+1}{x^2+1}\)

B = \(\dfrac{x^2+4x+4-x^2-1}{x^2+1}\)

B = \(\dfrac{\left(x+2\right)^2-\left(x^2+1\right)}{x^2+1}=\dfrac{\left(x+2\right)^2}{x^2+1}-1\ge-1\)

=> GTNN của B = -1 khi x + 2 = 0 <=> x = -2

Vậy B_min = -1 <=> x = -2

+) GTLN

B = \(\dfrac{4x+3}{x^2+1}\)

B = \(\dfrac{4x^2+4-4x^2-1+4x}{x^2+1}\)

B = \(\dfrac{4\left(x^2+1\right)-\left(2x-1\right)^2}{x^2+1}\)

B = \(4-\dfrac{\left(2x-1\right)^2}{x^2+1}\le4\)

=> GTLN của B = 4 <=> 2x - 1 = 0 <=> x = \(\dfrac{1}{2}\)

Vậy B_max = 4 <=> x = \(\dfrac{1}{2}\)