Shortest distance from origin to the curve y - e x - e - x -2A- -2B- 1 - 3 -2D- 4 -2

The correct option is B 1Let (x, y) is the point on the curve y=e^x+e^ x/2 nearer to origin ∴ op^2=x^2+y^2=x^2+ ( e^x+e^ x/2 )^2 Let f(x)=x^2+1/4(e^2x+e^ 2x+2) ...

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Byju's Answer

Standard XII

Mathematics

Normal of a Curve y = f(x)

Shortest dist...

Question

Shortest distance from origin to the curve $y = \frac{e^{x} + e^{- x}}{2}$

1

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\sqrt{2}

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3 \sqrt{2}

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4 \sqrt{2}

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Solution

The correct option is A $1$
Let (x, y) is the point on the curve $y = \frac{e^{x} + e^{- x}}{2}$ nearer to origin
$∴ o p^{2} = x^{2} + y^{2} = x^{2} + {(\frac{e^{x} + e^{- x}}{2})}^{2}$
$L e t f (x) = x^{2} + \frac{1}{4} (e^{2 x} + e^{- 2 x} + 2)$
$\Rightarrow f^{'} (x) = 2 x + \frac{1}{2} (e^{2 x} - e^{- 2 x})$
For maximum (or) minimum, we have $f^{'} (x) = 0 \Rightarrow e^{- 2 x} - e^{2 x} = 4 x$
$\Rightarrow x = 0 a l s o f^{''} (0) > 0$
$∴$ at x = 0 shortest distance exist i.e ‘1’

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Shortest distance from origin to the curve y=ex+e−x2

Shortest distance from origin to the curve $y = \frac{e^{x} + e^{- x}}{2}$