Answer :

Given,


The triangle is right angled triangle.


Hypotenuse is 5cm.



Let us consider,


The base of the triangle is ‘a’.


The adjacent side is ‘b’.


Now AC2 = AB2 + BC2


As AC = 5, AB = b and BC = a


25 = a2 + b2


b2 = 25 – a2 ---- (1)


Now, the area of the triangle is



Squaring on both sides



Substituting (1) in the area formula


----- (2)


For finding the maximum/ minimum of given function, we can find it by differentiating it with a and then equating it to zero. This is because if the function Z (x) has a maximum/minimum at a point c then Z’(c) = 0.


Differentiating the equation (2) with respect to a:




[Since ]


----- (3)


To find the critical point, we need to equate equation (3) to zero.




a=0 (or)



[as a cannot be zero]


Now to check if this critical point will determine the maximum area, we need to check with second differential which needs to be negative.


Consider differentiating the equation (3) with a:



----- (4)


[Since ]


Now let us find the value of



As , so the function A is maximum at


Substituting value of A in (1)




Now the maximum area is




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