Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2x^2 + (\cos \theta) x - 1 = 0, \theta \in (0, 2\pi)$. If $m$ and $M$ are the minimum and the maximum values of $\alpha_\theta^4 + \beta_\theta^4$, then $16(M + m)$ equals:

Step 1: Find sum and product of roots

For a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is $-b/a$ and the product of the roots is $c/a$. Here, $a=2$, $b=\cos \theta$, and $c=-1$. We use these formulas to find the sum and product of $\alpha_\theta$ and $\beta_\theta$.

Step 2: Express sum of fourth powers in terms of sum and product of roots

We want to find the value of $\alpha_\theta^4 + \beta_\theta^4$. We can express this in terms of $(\alpha_\theta^2 + \beta_\theta^2)$ and $(\alpha_\theta \beta_\theta)$. We also know that $\alpha_\theta^2 + \beta_\theta^2 = (\alpha_\theta + \beta_\theta)^2 - 2\alpha_\theta \beta_\theta$.

Step 3: Substitute values and simplify

Now we substitute the expressions for $(\alpha_\theta + \beta_\theta)$ and $(\alpha_\theta \beta_\theta)$ into the formula from the previous step. We then simplify the expression to get a function of $\cos \theta$.

Step 4: Further simplification

Continuing the simplification, we expand the terms and combine them. Let $y = \cos \theta$. Since $\theta \in (0, 2\pi)$, the range of $\cos \theta$ is $[-1, 1]$. The problem states that the roots are distinct, which means the discriminant must be positive. The discriminant is $b^2 - 4ac = (\cos \theta)^2 - 4(2)(-1) = \cos^2 \theta + 8$. Since $\cos^2 \theta \ge 0$, the discriminant is always positive, so the roots are always distinct for any $\theta$.

Step 5: Find minimum and maximum values

Let $f(y) = \frac{1}{16}y^4 + \frac{1}{2}y^2 + \frac{1}{8}$, where $y = \cos \theta$. Since $y^2$ is involved, we can let $z = y^2$. Then $z \in [0, 1]$. The function becomes $g(z) = \frac{1}{16}z^2 + \frac{1}{2}z + \frac{1}{8}$. This is a parabola opening upwards, so its minimum and maximum values on $[0, 1]$ occur at the endpoints. For $z=0$, $g(0) = \frac{1}{8}$. For $z=1$, $g(1) = \frac{1}{16} + \frac{1}{2} + \frac{1}{8} = \frac{1+8+2}{16} = \frac{11}{16}$. Thus, $m = \frac{1}{8}$ and $M = \frac{11}{16}$.

Step 6: Calculate $16(M+m)$

Finally, we substitute the values of $M$ and $m$ into the expression $16(M+m)$ and calculate the result.

Step 7: Final Calculation

We add the fractions inside the parenthesis and then multiply by 16 to get the final answer.

More from this chapter

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• Simplify (1 + ²)^5.
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