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

1 Answer
Dean R.
Apr 23, 2018

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#.

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