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burhan101
Determine the coordinates of the points on f(x)=x³+2x² where the tangent lines are perpendicular to y-x+2=0. How do I approach this question ?
:O why'd you delete it
So basically you have to find the coordinates of points on f(x) such that the derivative, or the slope of the tangent, is the negative reciprocal of the slope of the given line y = x - 2. This implies that the tangents at those coordinates on f(x) will be perpendicular to the line y = x - 2. So we know that the derivative must be the negative reciprocal of the slope of y = x - 2 at those coordinates so f'(x) must be -1 at those coordinates. Let's find f'(x): \[\bf f'(x)=3x^2+4x\] We know that the slope of the tangent at those coordinates of f(x) or the derivative is -1. So we equate f'(x) with -1 and solve for x:\[\bf 3x^2+4x=-1 \rightarrow 3x^2+4x+1=0 \rightarrow (3x+1)(x+1)=0 \implies x = -\frac{ 1 }{ 3 },-1\]Plug these x-values back in to f(x) to get the y-values and that will give you the coordinates. @burhan101
@genius12 thank you for the explanation ! very clear :)
How would I determine the equations for the tangent for f(x) ?
Why do you need the equation of the tangent lines at the coordinates? @burhan101
it's part b of the question
You have the coordinates right?
Yes \[(\frac{ -1 }{ x },\frac{ 5 }{ 27 })\] and (-1,1)
whats the x there for.
typo it was supposed to be a 3
Anyway, to find the equation of the tangent line at the coordinates, we need a point and a slope. We already know that since the tangent lines at these coordinates are perpendicular to y = x - 2, the slope for both tangents will be -1. We already have the points each tangent goes through. First one goes through (-1/3, 5/27) and second one goes through (-1, 1). The equation of tangent line will be int he form:\[\bf y= mx+b\]Where m is the slope, which is -1, and we need b. Since we know a point for each tangent, plug in the x-value and y-value of the coordinates and solve for b. This will then give you the slope-intercept equation of the tangents at the coordinates. @burhan101