Minimum distance between the curves f(x)=ex and g(x)= ln x is

Question

Minimum distance between the curves f(x)=ex and g(x)= ln x is (A) √2 (B) (1/√2) (C) 1 (D) 2(√2-1)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/d60b2722bdaf23bef03b503bef79efa9-.png" /> Minimum distance is the distance along the common normal to both the curves i.e. y= x must be parallel to the tangent as both the curves are inverse of each other. ( dy / dx )| x 1= e x 1=1 ⇒ x 1=0 text and y 1=1 ∴ AB =√2

Q. Minimum distance between the curves $f (x) = e^{x}$ and $g (x) = ln x$ is

$2$

$\frac{1}{2}$

1

$2 (2 - 1)$

Minimum distance is the distance along the common normal to both the curves i.e. $y =$ $x$ must be parallel to the tangent as both the curves are inverse of each other.
$\frac{d y}{d x} ∣_{x_{1}} = e^{x_{1}} = 1$
$\Rightarrow x_{1} = 0 and y_{1} = 1$
$∴ A B = 2$

Q. Minimum distance between the curves f(x)=ex and g(x)=lnx is

2​

2​1​

1

2(2​−1)

Minimum distance is the distance along the common normal to both the curves i.e. y= x must be parallel to the tangent as both the curves are inverse of each other. dxdy​∣x1​​=ex1​=1 ⇒x1​=0 and y1​=1 ∴AB=2​

Q. Minimum distance between the curves $f (x) = e^{x}$ and $g (x) = ln x$ is

$2$

$\frac{1}{2}$

$2 (2 - 1)$

Minimum distance is the distance along the common normal to both the curves i.e. $y =$ $x$ must be parallel to the tangent as both the curves are inverse of each other.
$\frac{d y}{d x} ∣_{x_{1}} = e^{x_{1}} = 1$
$\Rightarrow x_{1} = 0 and y_{1} = 1$
$∴ A B = 2$