The true set of real values of x for which the function, f(x)=x ln x-x+1 is positive is

Question

The true set of real values of x for which the function, f(x)=x ln x-x+1 is positive is (A) (1, ∞) (B) (1 / e, ∞) (C) [e, ∞) (D) (0,1) ∪(1, ∞)

Tardigrade Admin · Accepted Answer

f ( x )= x ln x - x +1 ⇒ f prime( x )=1+ ln x -1 text Hence f ( x ) text is uparrow text for x >1 text and f ( x ) text is downarrow text for 0< x <1 ⇒ f (1) text is the least value ∴ f ( x )> f (1) text for all x >1 text as well as 0< x <1 ∴ x ln x - x +1>0 text (f (1)=0) ⇒ text (D)

Q. The true set of real values of $x$ for which the function, $f (x) = x ln x - x + 1$ is positive is

$(1, \infty)$

$(1/ e, \infty)$

$[e, \infty)$

$(0, 1) \cup (1, \infty)$

$f (x) = x ln x - x + 1 \Rightarrow f^{'} (x) = 1 + ln x - 1$
$Hence f (x) is ↑ for x > 1 and f (x) is ↓ for 0 < x < 1$
$\Rightarrow f (1) is the least value$
$∴ f (x) > f (1) for all x > 1 as well as 0 < x < 1$
$∴ x ln x - x + 1 > 0 (f (1) = 0) \Rightarrow (D)$

Q. The true set of real values of x for which the function, f(x)=xlnx−x+1 is positive is

(1,∞)

(1/e,∞)

[e,∞)

(0,1)∪(1,∞)