What is the equation of the normal line of #f(x)=2x+3/x# at #x=1#?

1 Answer
Oct 31, 2016

# :. y=x+4#

Explanation:

# f(x) = 2x+3/x #
# :. f(x) = 2x+3x^-1 #

Differentiating wrt #x# gives us:
# f'(x) = 2+3(-x^-2) #
# :. f'(x) = 2-3/x^2 #

When # x=1 => f(1)=2+3 = 5 #
and, # f'(1) = 2-3=-1 #

So the gradient of the tangent at #x=1# is #m'=-1#, and the tangent and normal are perpendicular so the product of their gradients is #-1#
Hence, gradient of Normal is #m=-1/(-1)=1#

So the Normal has gradient #m=1# and it passes through #(1,5)#, so using # y-y_1=m(x-x_1) # the Normal equation is given by:

# y-5=(1)(x-1) #
# :. y=x+4#

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