Question

Se considera functia f(x)=e^x+e^1-x. Demonstrati ca 2sqrt(e) mai mic sau egal decat f(x) si mai mic sau egal decat 1+e

Answer 1

Trebuia specificat ca  0 <= x <= 1/2
f(x)=e^x + e^(1-x)  f' = e^x - *e^(1-x) = (e^2x - e)/e^x
f' = 0 => e^2x - e = 0 => e^2x = e^1 =>  2x = 1 => x = 1/2
Ptr  0 <= x <= 1/2  f' > 0 => f este descresc. => 1/2 abcisa pct. minim
f(1/2) = e^(1/2) + e^(1/2 = 2rad(e)  =  val. minima
f(0) = e^0 + e^1 = 1 + e = val maxima  
 2rad(e) <=  f(x) < = 1 + e