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three real and positive rootsthree real and negative rootsone real rootthree real roots such that sum of roots is zero

Solution :

Suppose degree of f(x) = n. Then degree of f'=n-1 and degree of `f''=n-2.` So, <br> `n=n-1+n-2` <br> Hence, n=3. <br> So, put `f(x)=ax^(3)+bx^(2)+cx+d (where a ne 0).` <br> From `f(2x)=f'(x).f''(x),` we have <br> `8ax^(3)+4bx^(2)+2cx+d=(3ax^(2)+2bx+c)(6ax+2b)` <br> `=18a^(2)x^(3)+18abx^(2)+(6ac+4b^(2))x+2bc` <br> Comparing coefficients of terms, we have <br> `18a^(2)=8a" "rArra=4//9` <br> `18ab=4b" "rArrb=0` <br> `2c=6ac+4b^(2)" "rArrc=0` <br> `d=2bc" "rArrd=0` <br> `"or "f(x)=(4x^(3))/(9),` which is clearly one-one and onto <br> `"or "f(3)=12` <br> `"Also, "(4x^(3))/(9)=xor x=0, x= pm 3//2.` <br> Hence, sum of roots of equation is zero.**Definition and geometric meaning.**

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