Consider the functions f1(x)=x, f2(x)=2+ log e x, x0. The graphs of the functions intersect.

Question

Consider the functions f1(x)=x, f2(x)=2+ log e x, x>0. The graphs of the functions intersect. (A) once in (0,1) but never in (1, ∞) (B) once in (0,1) and once in (e2, ∞) (C) once in (0,1) and once in (e, e2) (D) more than twice in (0, ∞)

Tardigrade Admin · Accepted Answer

f1(x)=x, f2(x)=2+ log e x Let g(x)=f2(x)-f1(x)=2+ log e x-x g(0+)<0, g(1)>0, g(e)>0, g(e2)<0 and value of g(x) for all x ≥ e2 is negative. ∴ g(x)=0 has two roots in (0,1) and (e, e2)

Q. Consider the functions $f_{1} (x) = x, f_{2} (x) = 2 + lo g_{e} x, x > 0$ . The graphs of the functions intersect.

once in $(0, 1)$ but never in $(1, \infty)$

once in $(0, 1)$ and once in $(e^{2}, \infty)$

once in $(0, 1)$ and once in $(e, e^{2})$

more than twice in $(0, \infty)$

$f_{1} (x) = x, f_{2} (x) = 2 + lo g_{e} x$
Let $g (x) = f_{2} (x) - f_{1} (x) = 2 + lo g_{e} x - x$
$g (0^{+}) < 0, g (1) > 0, g (e) > 0, g (e^{2}) < 0$
and value of $g (x)$ for all $x \geq e^{2}$ is negative.
$∴ g (x) = 0$ has two roots in $(0, 1)$ and $(e, e^{2})$

Q. Consider the functions f1​(x)=x,f2​(x)=2+loge​x,x>0. The graphs of the functions intersect.

once in (0,1) but never in (1,∞)

once in (0,1) and once in (e2,∞)

once in (0,1) and once in (e,e2)

more than twice in (0,∞)