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 - Maths - Application of Derivatives

Question

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 Regards

Jasleen Kaur · Accepted Answer

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]

Regards