Question

Find the equation of the tangent to the curve y= 2-√x perpendicular to the straight line y+4x-4=0 ?

Answer 1

The slope of the perpendicular is `1/4`, but the derivative of the curve is `-1/{2sqrt{x}}`, which will always be negative, so the tangent to the curve is never perpendicular to `y+4x=4`.

Explanation:

`f(x) = 2 - x^{1/2}`

`f'(x) = - 1/2 x^{-1/2} = -1/{2sqrt{x}}`

The line given is

`y = -4x + 4`

so has slope `-4`, so its perpendiculars have the negative reciprocal slope, `1/4`. We set the derivative equal to that and solve:

`1/4 = -1/{2 sqrt{x} }`

`sqrt{x} = -2`

There's no real `x` that satisfies that, so no place on the curve where the tangent is perpendicular to `y+4x=4`.