amy0799
  • amy0799
f(x)=(x+8)^2+3 Find the point on the graph of function that is closest to the point (-2, 6).
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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DanJS
  • DanJS
both the given point, and th epoint on the function would contain the same line, Normal to the curve right
baru
  • baru
i think you will have to write another function for distance distance between f(x) and given point \[\sqrt{(x+2)^2+(f(x)-6)^2}\]
DanJS
  • DanJS
yes, the direction should be normal to the curve,

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baru
  • baru
we can square the expression, because min will occur when the expression under the square root is minimum
DanJS
  • DanJS
yeah minimize the distance
baru
  • baru
\[d={(x+2)^2+(f(x)-6)^2}\]
baru
  • baru
just set the derivative of that=0
DanJS
  • DanJS
they give you the function y = place tha ty value in to the distance function , to get distance in terms of only x
DanJS
  • DanJS
if you doin algebra
amy0799
  • amy0799
f\[f(x)=x^4+32x^3+379x^2+1956x+3725\]
amy0799
  • amy0799
Is that correct?
baru
  • baru
yea, i guess, dont call it f(x) you might confuse yourself, call it s(x) or something else
DanJS
  • DanJS
\[d = \sqrt{(x+2)^2+(f(x)-6)^2}\] from above replace f(x), for the function value, f(x) =(x+8)2 + 3
DanJS
  • DanJS
simplify all that, so you have distance d in terms of just x,
DanJS
  • DanJS
minimize that function,
baru
  • baru
you can minimiz d^2 instead of d \[d^2=x^4+32x^3+379x^2+1956x+3725\] is what amy is getting
baru
  • baru
differentiate, equate to zero, solve and find x, and corresponding f(x)
baru
  • baru
you might need a calculator :P
amy0799
  • amy0799
I want to make sure the function i got is correct first, is it?
baru
  • baru
or @DanJS has got me thinking, at the minimum point, the tangent to the curve will be perpendicular to the line connecting point on curve and (-2,6)
baru
  • baru
yea, ur function looks correct
DanJS
  • DanJS
yeah , use the tangent at that point (x,y) and the normal line will have slope -1/y'
DanJS
  • DanJS
so just take the derivative, f'(x), that is the tangent slope at the point (x,y) the slope of the line between (-2,6) and (x,y) on the function, has a slope perpendicular to the tangent slope normal to curve slope = -1/y'
amy0799
  • amy0799
\[s'(x)=-1/4x^3+96x^2+758x+1956\]
DanJS
  • DanJS
by the way , i just put that distance function d(x) into the calc, and found the d ' (x), and solved that for being 0, x=6
DanJS
  • DanJS
back to the derivative thing slope of normal line is -1/y ' or -1/(2x + 16 and the components of that slope in x and y directions, forming the right triangle between the points are y-6 and x+2 the slope of that normal -1/(2x+16) = (y - 6) / x + 2)
baru
  • baru
How did u get -1/4
amy0799
  • amy0799
how did you get 2x+16?
DanJS
  • DanJS
the derivative of the function y ' = 2(x+8)^1*1 = 2x + 16
amy0799
  • amy0799
i thought you're taking the derivative of (x+2)^2+((x+8)^2+3-6)^2
baru
  • baru
\[s(x)=x^4+32x^3+379x^2+1956x+3725\] just take the derivative of this; other stuff was just an alternate method
DanJS
  • DanJS
that is the slope of the tangent at a point on the function the slope of the line between that point (x,y) and (-2,6) will be perpendicular to that tangent line -1/y ' or -1/(2x+16)
baru
  • baru
\[s'(x)=-1/4x^3+96x^2+758x+1956\] this is what you got right?
baru
  • baru
how did you get the -1/4 at the start?
DanJS
  • DanJS
that should be the same as change in y over change in x. or (y - 6 ) / (x + 2)
DanJS
  • DanJS
\[\large \frac{ -1 }{ 2x+16 } = \frac{ y - 6 }{ x + 2 }\] replace the function value for the y variable
DanJS
  • DanJS
\[\large \frac{ -1 }{ 2x+16 } = \frac{ [(x+8)^2 + 3] - 6 }{ x + 2 }\] that solves to x = -6 again, same as the distance formula wau
DanJS
  • DanJS
point on the curve (x,y) = (-6 , f(-6))
amy0799
  • amy0799
so plug -6 into the original function?
amy0799
  • amy0799
i got the answer, thank you for the help
DanJS
  • DanJS
the derivative of that distance, is a cubic function right?, that is hard to solve , more than just doing -1/y'(x) = (y - y1) / (x - x1)
amy0799
  • amy0799
I understand now. (-6,7)

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