Question

Question
​
Two sides of a triangle are given find the angle between them such that the area of the triangle is maximum.

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Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that

The two sides of a triangle ABC be represented as a and b

and

Let x be the angle included between the two sides a and b.

Then area of triangle ABC is given by

$\rm :\longmapsto\:Area_{\triangle ABC}, \: A \: = \dfrac{1}{2}ab \: sinx$

[ For, proof of the above result, see the attachment ]

Now, Differentiate both sides w. r. t. x, we get

$\rm :\longmapsto\:\dfrac{d}{dx} A \: = \dfrac{d}{dx}\dfrac{1}{2}ab \: sinx$

$\rm :\longmapsto\:\dfrac{dA}{dx} \: = \dfrac{1}{2}ab \:\dfrac{d}{dx} sinx$

$\rm :\longmapsto\:\dfrac{dA}{dx} \: = \dfrac{1}{2}ab \:cosx - - - (1)$

$\red{\bigg \{ \because \: \dfrac{d}{dx}sinx = cosx\bigg \}}$

For maxima or minima,

$\rm :\longmapsto\:\dfrac{dA}{dx} \: =0$

$\rm :\longmapsto\:\dfrac{1}{2}ab \: cosx \: = \: 0$

$\rm :\longmapsto\: \: cosx \: = \: 0$

$\bf\implies \:x = \dfrac{\pi}{2}$

Now, from equation (1), we have

$\rm :\longmapsto\:\dfrac{dA}{dx} \: = \dfrac{1}{2}ab \:cosx$

On differentiating both sides w. r. t. x, we get

$\rm :\longmapsto\:\dfrac{d}{dx}\dfrac{dA}{dx} \: = \dfrac{d}{dx}\dfrac{1}{2}ab \:cosx$

$\rm :\longmapsto\:\dfrac{ {d}^{2} A}{d {x}^{2} } \: = \dfrac{1}{2}ab \dfrac{d}{dx}\:cosx$

$\rm :\longmapsto\:\dfrac{ {d}^{2} A}{d {x}^{2} } \: = - \: \dfrac{1}{2}ab \: sinx$

$\red{\bigg \{ \because \: \dfrac{d}{dx}cosx = - \: sinx\bigg \}}$

Now,

$\rm :\longmapsto\:\dfrac{ {d}^{2} A}{d {x}^{2} }_{at \: x \: = \dfrac{\pi}{2}} \: = - \: \dfrac{1}{2}ab \: sin\dfrac{\pi}{2}$

$\rm :\longmapsto\:\dfrac{ {d}^{2} A}{d {x}^{2} }_{at \: x \: = \dfrac{\pi}{2}} \: = - \: \dfrac{1}{2}ab \: \: \: \: \: \: \: \: \: \: \: \: \: \{ \because \: sin\dfrac{\pi}{2} = 1 \}$

$\rm :\longmapsto\:\dfrac{ {d}^{2} A}{d {x}^{2} }_{at \: x \: = \dfrac{\pi}{2}} \: < 0$

$\bf :\longmapsto\:Area_{\triangle ABC} \: is \: maximum \: when\: x \: = \dfrac{\pi}{2}$

Basic Concept Used :-

Let y = f(x) be a given function.

To find the maximum and minimum value, the following steps are follow :

1. Differentiate the given function.

2. For maxima or minima, put f'(x) = 0 and find critical points.

3. Then find the second derivative, i.e. f''(x).

4. Apply the critical points ( evaluated in second step ) in the second derivative.

5. Condition :-

The function f (x) is maximum when f''(x) < 0.

The function f (x) is minimum when f''(x) > 0.