Q. Let S be the non empty set containing all 'a' for which f(x)=(4a-7)/3x3+(a-3)x2+x+5 is monotonic for all x is a element of R. Find S.

The correct answer is 'a' is a element of [2,8]
but my answer is  a<2 and a>8

I first find f'(x) which comes out to  be =(4a-7)x2+2(a-3)x+1
Then I take the discriminant>0
which gives me the answer ​a<2 and a>8
but if i take the discriminant<0 then i get the correct answer.

Can you please explain why we should take the discriminant<0 or if i am making another mistake please tell.



Dear Student,

We have,
f(x)=(4a-7)3x3+(a-3)x2+x+5f'(x)=(4a-7)x2+2(a-3)+1D=(2(a-3))2-4(4a-7)(1)=4(a2-6a+9)-16a+28=4a2-24a+36-16a+28=4a2-40a+64=4(a2-10a+16)=4(a2-2a-8a+16)=4(a(a-2)-8(a-2))=4(a-2)(a-8)Now, for f(x)to be monotonic, f'(x)>0Therefore, D<0 to make f'(x)>0.So, take determinant less then zero and you will get your required resulti.e. a[2,8]


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