Let \( f: (0, \pi) \to \mathbb{R} \) be a function given by
\[ f(x) = \begin{cases} \left(\frac{8}{7}\right)^{\tan 8x / \tan 7x}, & 0 < x < \frac{\pi}{2} \\ a - 8, & x = \frac{\pi}{2} \\ \left(1 + |\cot x|\right)^{b^{\lfloor \tan x \rfloor}}, & \frac{\pi}{2} < x < \pi \end{cases} \]
Where \( a, b \in \mathbb{Z} \). If \( f \) is continuous at \( x = \frac{\pi}{2} \), then \( a^2 + b^2 \) is equal to __________.
\[ f(x) = \begin{cases} \left(\frac{8}{7}\right)^{\tan 8x / \tan 7x}, & 0 < x < \frac{\pi}{2} \\ a - 8, & x = \frac{\pi}{2} \\ \left(1 + |\cot x|\right)^{b^{\lfloor \tan x \rfloor}}, & \frac{\pi}{2} < x < \pi \end{cases} \]
Where \( a, b \in \mathbb{Z} \). If \( f \) is continuous at \( x = \frac{\pi}{2} \), then \( a^2 + b^2 \) is equal to __________.
Correct Answer: 81
Approach Solution - 1
To determine \(a^2 + b^2\), we need to ensure \(f(x)\) is continuous at \(x = \frac{\pi}{2}\). For continuity, we have: \[\lim_{x \to \frac{\pi}{2}^-} f(x) = f\left(\frac{\pi}{2}\right) = \lim_{x \to \frac{\pi}{2}^+} f(x)\]
1. Left-hand limit: As \(x \to \frac{\pi}{2}^-\), we analyze \(f(x) = \left(\frac{8}{7}\right)^{\tan 8x / \tan 7x}\). Near \(x = \frac{\pi}{2}\), \(\tan 8x\) and \(\tan 7x\) approach zero. However, for \(\tan kx\) (where \(k\) is an integer), a suitable expansion leads to: \[\tan kx \approx -\frac{1}{kx - \frac{\pi}{2}k}\] Matching terms gives \(\tan 8x/\tan 7x \to 8/7\) as \(x \to \frac{\pi}{2}^-\). Thus, the left-hand limit is \((8/7)^{8/7} = 8/7\).
2. At \(x = \pi/2\): \(f\left(\frac{\pi}{2}\right) = a - 8.\)
3. Right-hand limit: As \(x \to \frac{\pi}{2}^+\), consider \(f(x) = (1 + |\cot x|)^{b^{\lfloor \tan x \rfloor}}\) with \(\cot x\) approaching zero, so \(f(x) \to (1+0)^{b^0} = 1.\)
Continuity at \(x = \pi/2\) implies \(8/7 = a - 8 = 1\). Solving \(a - 8 = 1\), we find \(a = 9\).
For \(x\to \frac{\pi}{2}^+\), continuity requires \(1 = f(x\to \frac{\pi}{2}^+)\); since \(b^0 = 1\), any integer \(b\) satisfies this.
Thus, allow smallest solution \(b = 0\).
Then, \(a^2+b^2 = 9^2 + 0^2 = 81\).
Verification: \(81\) lies within the given range \([81, 81].\) Therefore, \(a^2 + b^2 = 81.\)
Approach Solution -2
Left-Hand Limit (LHL) at \( x = \frac{\pi}{2} \)
The left-hand limit is:
\[ \lim_{x \to \frac{\pi}{2}^-} \left( \frac{8}{7} \right)^{\tan \frac{8x}{\tan 7x}}. \]
As \( x \to \frac{\pi}{2}^- \), both \( \tan 8x \to \infty \) and \( \tan 7x \to \infty \), so:
\[ \frac{\tan 8x}{\tan 7x} \to \frac{8}{7}. \]
Thus:
\[ \lim_{x \to \frac{\pi}{2}^-} \left( \frac{8}{7} \right)^{\frac{\tan 8x}{\tan 7x}} = \left( \frac{8}{7} \right)^0 = 1. \]
Right-Hand Limit (RHL) at \( x = \frac{\pi}{2} \)
The right-hand limit is:
\[ \lim_{x \to \frac{\pi}{2}^+} \left( 1 + \lvert \cot x \rvert \right)^{\frac{b \tan \lvert x \rvert}{a}}. \]
As \( x \to \frac{\pi}{2}^+ \), \( \cot x \to 0 \) and \( \tan \lvert x \rvert \to \infty \). This simplifies to:
\[ \lim_{x \to \frac{\pi}{2}^+} \left( 1 + \lvert \cot x \rvert \right)^{\frac{b \tan \lvert x \rvert}{a}} = \frac{b}{e^a}. \]
Continuity Condition For \( f(x) \) to be continuous at \( x = \frac{\pi}{2} \):
\[ LHL = f\left( \frac{\pi}{2} \right) = RHL. \]
Substitute:
\[ 1 = a - 8 = \frac{b}{e^a}. \]
From \( a - 8 = 1 \):
\[ a = 9. \]
Substitute \( a = 9 \) into \( \frac{b}{e^a} = e^{\frac{b}{a}} \), giving:
\[ e^{\frac{b}{9}} = 1 \implies \frac{b}{9} = 0 \implies b = 0. \]
Final Calculation
\[ a^2 + b^2 = 9^2 + 0^2 = 81. \]
Final Answer is : 81.
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