AP EAMCET Engineering 21 Apr 2019 Shift 2 Paper

Consider
f(x)=ax2+bx+c
f(2)=4a+2b+c=2‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(I)
f(3)=9a+3b+c=5‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(II)
f(4)=16a+4b+c=10‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(III)
Subtract equation (I) from equation (II),
5a+b=3‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(IV)
Subtract equation (II) from equation (III),
7a+b=5‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(V)
Subtract equation (IV) from equation (V),
2a=2
a=1
Substitute the value in equation (IV),
5×1+b=3
b=−2
Now,
f′(x)=2x−2
f′′(x)=2
since, f′′(x)>1 for some x∈(2,4)