Consider the curve #x^2 + 2x + y^4 + 4y = 5#. What is the sum of the x-coordinates of the two points on the curve where the line tangent to the curve is vertical?

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Answer 1 · 2015-11-19T20:10:52

A vertical tangent line has an undefined ( infinite ) slope. Solve the first derivative such that #y'# equals infinity.

Using implicit differentiation :

#2x+2+4y^3y'+4y'=0#

Next, solve for #y'#

#y'=(x+1)/[2(y^3+1)]#

Now, #y'# will equal infinity when the denominator equals zero ...

#2(y^3+1)=0#, so solving for y ...

#y=-1#

Inserting #y=-1# back into the original equation, solve for x ...

#x=-4# and #x=-2#

Finally, the coordinates of the vertical tangent lines are:

#(-4,-1) and (2,-1)#

Sum of the x-coordinates #=-4+2=-2#

hope that helped

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