Question

For the functions $f(\theta) = \alpha \tan^{2}\theta + \beta \cot^{2}\theta$, and $g(\theta) = \alpha \sin^{2}\theta + \beta \cos^{2}\theta$, $\alpha>\beta>0$, let $\min_{0<\theta<\frac{\pi}{2}} f(\theta) = \max_{0<\theta<\pi} g(\theta)$. If the first term of a G.P. is $\left( \frac{\alpha}{2\beta} \right)$, its common ratio is $\left( \frac{2\beta}{\alpha} \right)$ and the sum of its first 10 terms is $\frac{m}{n}$, $\gcd(m, n) = 1$, then $m + n$ is equal to ________.

Hint:

Apply the AM-GM inequality to find the minimum of f(theta) and analyze the range of sin^2(theta) to find the maximum of g(theta). Then use the GP sum formula.

Answer 1

Correct Answer: 1279

We can also determine the extremum values using calculus and then solve for the sum of the series.

Step 1: Finding extrema using derivatives.
For $f(\theta) = \alpha \tan^2 \theta + \beta \cot^2 \theta$, the derivative is:
$f'(\theta) = 2\alpha \tan \theta \sec^2 \theta - 2\beta \cot \theta \csc^2 \theta$.
Setting $f'(\theta) = 0$ for a minimum:
$\alpha \frac{\sin \theta}{\cos^3 \theta} = \beta \frac{\cos \theta}{\sin^3 \theta} \implies \tan^4 \theta = \frac{\beta}{\alpha} \implies \tan^2 \theta = \sqrt{\frac{\beta}{\alpha}}$.
Substituting this back into $f(\theta)$:
$f_{min} = \alpha \sqrt{\frac{\beta}{\alpha}} + \beta \sqrt{\frac{\alpha}{\beta}} = \sqrt{\alpha \beta} + \sqrt{\alpha \beta} = 2\sqrt{\alpha \beta}$.

Step 2: For $g(\theta) = \alpha \sin^2 \theta + \beta \cos^2 \theta$, the derivative is:
$g'(\theta) = 2\alpha \sin \theta \cos \theta - 2\beta \sin \theta \cos \theta = (\alpha - \beta) \sin 2\theta$.
In $(0, \pi)$, $\sin 2\theta = 0$ at $\theta = \pi/2$.
Since $\alpha>\beta$, the second derivative $g''(\theta) = 2(\alpha - \beta) \cos 2\theta$ is negative at $\theta = \pi/2$ (giving $g'' = -2(\alpha-\beta)$), so it is a maximum.
$g_{max} = \alpha \sin^2(\pi/2) + \beta \cos^2(\pi/2) = \alpha$.

Step 3: Finding the relationship.
Equating $2\sqrt{\alpha \beta} = \alpha$ lead to $\alpha^2 = 4\alpha \beta$, hence $\alpha = 4\beta$.
The first term of the G.P. is $a = \frac{\alpha}{2\beta} = 2$.
The common ratio is $r = \frac{2\beta}{\alpha} = \frac{2\beta}{4\beta} = \frac{1}{2}$.

Step 4: Summation.
The sum of first 10 terms is:
$$S_{10} = \sum_{k=0}^{9} 2 \cdot (1/2)^k = 2 \left( \frac{1 - (1/2)^{10}}{1 - 1/2} \right) = 4 \left( 1 - \frac{1}{1024} \right) = \frac{1023}{256}$$
Comparing with $m/n$, we find $m = 1023$ and $n = 256$.
$m + n = 1023 + 256 = 1279$.