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प्रश्न
Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\]. What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.
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उत्तर
\[\text { As, the area of the triangle, A } = \frac{1}{2}ab\sin\theta\]
\[ \Rightarrow A\left( \theta \right) = \frac{1}{2}ab\sin\theta\]
\[ \Rightarrow A'\left( \theta \right) = \frac{1}{2}\text { ab }\cos\theta\]
\[\text { For maxima or minima, A}'\left( \theta \right) = 0\]
\[ \Rightarrow \frac{1}{2}ab\cos\theta = 0\]
\[ \Rightarrow \cos\theta = 0\]
\[ \Rightarrow \theta = \frac{\pi}{2}\]
\[\text { Also, A }''\left( \theta \right) = - \frac{1}{2}ab\sin\theta\]
\[\text { or,} A''\left( \frac{\pi}{2} \right) = - \frac{1}{2}ab\sin\frac{\pi}{2} = - \frac{1}{2}ab < 0\]
\[\text { i . e } . \theta = \frac{\pi}{2} \text { is point of maxima }\]
\[\text { Now }, \]
\[\text { The maximum area of the triangle } = \frac{1}{2}ab\sin\frac{\pi}{2} = \frac{ab}{2}\]
