Question

tìm GTNN:

A= x^4 + 3x^2 -2

B= (3x+1)^2 +(y-5)^2 -15

giúp mình nha

Answer 1

\(A=x^4+3x^2-2\)

\(=x^4+3x^2+\dfrac{9}{4}-\dfrac{17}{4}\)

\(=\left(x^2+\dfrac{3}{2}\right)^2-\dfrac{17}{4}\)

Ta có: \(\left(x^2+\dfrac{3}{2}\right)^2\ge0\forall x\Rightarrow\left(x^2+\dfrac{3}{2}\right)^2-\dfrac{17}{4}\ge-\dfrac{17}{4}\forall x\)

Dấu "=" xảy ra khi \(x^2+\dfrac{3}{2}=0\Leftrightarrow x^2=-\dfrac{3}{2}\)(vô lí)

Vậy A không có GTNN.

\(B=\left(3x+1\right)^2+\left(y-5\right)^2-15\)

Ta có: \(\left(3x+1\right)^2+\left(y-5\right)^2\ge0\forall x\Rightarrow\left(3x+1\right)^2+\left(y-5\right)^2-15\ge-15\forall x\)

Dấu "=" xảy ra khi \(3x+1=0\) và \(y-5=0\)

\(\Leftrightarrow x=-\dfrac{1}{3};y=5\)

Vậy MIN_B = -15 khi x = \(-\dfrac{1}{3}\) và y = 5.