6. - / 1 points\nfind the point on the line ( y = 3x + 4 ) that is closest to the origin.

Explanation:

Step1: Use the distance formula

The distance (d) between a point ((x,y)) and the origin ((0,0)) is (d=\sqrt{(x - 0)^2+(y - 0)^2}=\sqrt{x^{2}+y^{2}}). Since (y = 3x+4), substitute (y) into the distance formula: (d=\sqrt{x^{2}+(3x + 4)^{2}}=\sqrt{x^{2}+9x^{2}+24x + 16}=\sqrt{10x^{2}+24x + 16}). To simplify calculations, we can minimize the function (f(x)=10x^{2}+24x + 16) (because minimizing (f(x)) is equivalent to minimizing (d) as (d=\sqrt{f(x)}) and the square - root function is a monotonic increasing function for non - negative values).

Step2: Find the derivative of (f(x))

Using the power rule ((x^{n})^\prime=nx^{n - 1}), the derivative (f^\prime(x)=(10x^{2}+24x + 16)^\prime=20x+24).

Step3: Set the derivative equal to zero

Set (f^\prime(x)=0), so (20x+24 = 0). Solve for (x): [ \begin{align*} 20x&=-24\ x&=-\frac{24}{20}=-\frac{6}{5} \end{align*} ]

Step4: Find the value of (y)

Substitute (x =-\frac{6}{5}) into the equation (y = 3x+4). Then (y=3\times(-\frac{6}{5})+4=-\frac{18}{5}+\frac{20}{5}=\frac{2}{5})

Answer:

(\left(-\frac{6}{5},\frac{2}{5}\right))