Find the point on a parabola that is closest to a given point

Question

As you can see I found the equation but I don't know how to find the points. As far as I tried was $(7, 49)$ but it was wrong.

Answer 1

Minimizing the function is the same as minimizing its square. You wrote the equation for the distance, but the exercise asks its square, so we would have $$s(x) = (x - 7)^2 + (y - 0)^2$$ But the point $(x,y)$ is in the parabola $y = x^2$. So we get $$s(x) = (x - 7)^2 + x^4$$

Now, can you solve $s '(x) = 0 $? This will give you the critical points of $s$.

Answer 2

To find the minimum you need to compute the first derivative of $S(x)$ and find the critical points. We have

$$S'(x)=(4x^3+2x-14)\frac{1}{2\sqrt[]{(x-7)^2+x^4}}=0$$

$$4x^3+2x-14=0$$

This has one real solution: $x\approx1.4087$. See below

http://www.wolframalpha.com/input/?i=zeros+of+4x%5E3%2B2x-14

We know that $S$ must have a local minimum at that point because inspecting the plot of $S'(x)$ shows that to the left of this point the derivative is negative, and to the right the derivative is positive. See below

http://www.wolframalpha.com/input/?i=+%284x%5E3%2B2x-14%29%2F%282sqrt%28%28x-7%29%5E2%2Bx%5E4%29%29+at+x%3D1.4087

Answer 3

I do not know if you heard yet about the solution of cubic polynomials. A classical method is Cardano's which, for the equation $$4x^3+2x-14=0$$ will show that there is only one real root (the two other are complex conjugated) the exact expression being given by $$x=\frac{\sqrt[3]{63+5 \sqrt{159}}}{6^{2/3}}-\frac{1}{\sqrt[3]{6 \left(63+5 \sqrt{159}\right)}} \simeq 1.40872718612$$