Question

the least value of 5^x+5^-x is​

Answer 1

least power can be 0

it will bring the lanswer to 2

Answer 2

[math]\text{Let,} \quad y = \frac{x^2 -6x+5}{x^2 +2x+1}[/math]

[math]\implies x^2 y+2xy+y = x^2 -6x+5[/math]

[math]\implies (y-1)x^2 +2(y+3)x+(y-5) = 0 \quad ....(1)[/math]

(1) is a quadratic equation in x. Since, [math]x \in \mathbb{R}.[/math]

Therefore, discriminant[math](∆)[/math] of (1) must be non-negative.

i.e., [math]∆ \geq 0[/math]

[math]\implies 4(y+3)^2 -4(y-1)(y-5) \geq 0[/math]

[math]\implies y^2 +6y+9-y^2 +6y-5 \geq 0[/math]

[math]\implies 12y \geq -4[/math]

[math]\boxed{\therefore y \geq -\frac{1}{3}.}[/math]

Thus, least value of the given expression(y) is [math]\left(-\frac{1}{3}\right).[/math]

Hope, it'll help..!!

P.S.- Feel free to comment if you have any query with it.

Thank You!