Question

Find the largest natural number 'a' for which the maximum value of f(x)=a-1 + 2x - x^{​​​​​​2} is smaller than the minimum value of g(x)= x^{​​​​​​2} -2ax +10-2a.

Answer 1

Solution :-

We are given f(x) = a- 1 + 2x - x^2

= a - ( x^2 - 2x + 1 )

f(x) = a - ( x - 1) ^2

Maximum value of f(x) = a when x - 1 = 0 or x = 1

Now g(x) = x ^2 - 2ax + 10 - 2a


making the perfect square

= x ^2 - 2ax + a^2 - a ^2 +10 - 2a

= (x^2 - 2ax + a^2) + 10 - 2a -a^2

= ( x-a ) ^2 + 10 - 2a -a ^2

So minimum value of g(x) = 10 - 2a - a^2 when (x- a) = 0 or x = a


Now find maxium f(x) smaller than the minimum value of g(x)

a <10 - 2a - a^2


=> a ^2 + 3a - 10 <0
=> a^2 + 5a - 2a - 10 <0

= a(a + 5) - 2 ( a+ 5) < 0

( a - 2) ( a+5) <0

a = 2 , a = - 5

a belongs to ( -5, 2)

So greatest natural number a = 1 Answer