Posted by **Son** on Wednesday, December 7, 2011 at 11:21pm.

Find the point on the graph of the function closest to the given point. Function f(x) = xsquared Point (2, 1/2)

- math -
**Steve**, Thursday, December 8, 2011 at 11:54am
using distance from (2,1/2) to y=x^2,

d^2 = (x-2)^2 + (y-1/2)^2

= (x-2)^2 + (x^2 - 1/2)^2

= x^2 - 4x + 4 + x^4 - x^2 + 1/4

= x^4 - 4x + 17/4

d = sqrt(x^4 - 4x + 17/4)

d' = 2(x^3 - 1)/sqrt(x^4 - 4x + 17/4)

d' = 0 when x = 1

so, (2,1/2) is closest to (1,1)

______________________

using normal line, we want the line from (2,1/2) normal to the curve. The distance from P to the curve will be minimum along the line normal to the curve.

At (x,y) the slope of x^2 = 2x

so, the normal has slope -1/2x

So, the equation of the normal line from (2,1/2) is

(y - 1/2)/(x-2) = -1/2x

2x(x^2 - 1/2) = -(x-2)

2x^3 - x = -x + 2

2x^3 = 2

x = 1

So, the normal line from (2,1/2) is the line to (1,1)

## Answer this Question

## Related Questions

Math - Find the point on the graph of the function that is closest to the given ...

calculus - optimization find the point on the graph of the function that is ...

calc - let f be function given by f(x)= Ln(x)/x for all x> 0. the dervative ...

Calculus - find the point on the graph of the function that is closest to the ...

calculus - 8. Find the derivative of the function f(x) = cos^2(x) + tan^2(x) 9...

calculus - 1. Which of the following expressions is the definition of the ...

Calculus - Find the point P on the graph of the function y=\sqrt{x} closest to ...

Calculus - Find the point on the graph of function that is closest to the point...

College Algebra - If the point P is on the graph of a function f, find the ...

calculus - let f be the function f(x) = x^3 + 3x^2 - x + 2 a. the tangent to the...