Question

an investor wants to distribute funds between two investments to minimize risk. if x is the portion invested in the first option, and the risk function is $r(x) = 0.067x^2 - 0.08x + 0.04$, how should the investor distribute the funds? give your answers in decimal form rounded to the nearest thousandth. enter your answers in the boxes. first option: second option:

Explanation:

Step1: Identify function type

The risk function $R(x)=0.067x^2 - 0.08x + 0.04$ is a quadratic function in the form $ax^2+bx+c$ with $a>0$, so its minimum is at the vertex.

Step2: Calculate vertex x-value

The x-coordinate of the vertex of $ax^2+bx+c$ is $x=-\frac{b}{2a}$. Substitute $a=0.067$, $b=-0.08$:
$$x=-\frac{-0.08}{2\times0.067}=\frac{0.08}{0.134}\approx0.597$$

Step3: Find second portion

The total portion is 1, so the portion for the second option is $1 - x$.
$$1 - 0.597=0.403$$

Answer:

First option: $0.597$
Second option: $0.403$