Name: Local Maxima and Minima of a Function
Uploaded: 2019-07-19
Duration: 14 min 17 s
Description: Do you know what is minima or maxima of a function? This video also explains how to find local maxima and minima by differentiation in a detailed manner.

Question

Find the maximum perimeter of a triangle on a given base '

a

' and having the given vertical angle

α

.

Accepted Answer

We have,

By sine rule, we have

$\frac{a}{\sin α} = \frac{t_{1}}{\sin θ} = \frac{t_{2}}{\sin [π - (θ + α)]}$

$\Rightarrow \frac{a}{\sin α} = \frac{t_{1}}{\sin θ} = \frac{t_{2}}{\sin (θ + α)}$

$∴ t_{1} = (\frac{a}{\sin α}) \sin θ$ and $t_{2} = (\frac{a}{\sin α}) \sin (θ + α)$

To maximise perimeter, we must maximise $t_{1} + t_{2}$ . Let

$t_{1} + t_{2} = f (θ) = \frac{a}{\sin α} [\sin (θ) + \sin (θ + α)]$

$\Rightarrow f (θ) = \frac{2 a}{\sin α} \{\sin (θ + \frac{α}{2}) \cos (\frac{α}{2})\}$

$\Rightarrow f (θ) = \frac{2 a}{2 \sin (\frac{α}{2}) \cos (\frac{α}{2})} \{\sin (θ + \frac{α}{2}) \cos (\frac{α}{2})\}$

$\Rightarrow f (θ) = \frac{a}{\sin (\frac{α}{2})} [\sin (θ + \frac{α}{2})]$

Differentiating both sides w.r.t. $θ$ , we get

$f^{'} (θ) = a cosec (\frac{α}{2}) \cos (θ + \frac{α}{2})$

$f^{''} (θ) = - a cosec (\frac{α}{2}) \sin (θ + \frac{α}{2})$

For maximum perimeter,

$f^{'} (θ) = 0$

$\Rightarrow a cosec (\frac{α}{2}) \cos (θ + \frac{α}{2}) = 0$

$\Rightarrow \cos (θ + \frac{α}{2}) = 0$

$\Rightarrow θ + \frac{α}{2} = \frac{π}{2}$

$\Rightarrow θ = (\frac{π - α}{2})$

And, $f^{''} (θ) < 0$ , for $θ = (\frac{π - α}{2})$

Hence, maximum perimeter is

$= a + t_{1} + t_{2}$

$= a (1 + cosec (\frac{α}{2}))$

Find the maximum perimeter of a triangle on a given base ' $a$ ' and having the given vertical angle $α$ .

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Find the maximum perimeter of a triangle on a given base 'a' and having the given vertical angle α.

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Find the maximum perimeter of a triangle on a given base ' $a$ ' and having the given vertical angle $α$ .