Find the sum of all possible integral value(s) of 'a' for which F(x)= x^3 3 +(a-3) x^2+x-13 has negative point of local minimum in the interval [1,100] .

Step-by-step solution

We have F(x)= x^3 3 +(a-3) x^2+x-13 For F(x) to have negative point of local minimum, the equation F^ (x)=0 must have two distinct negative roots. Now, F ^ ( x )= x ^2+2(a-3) x +1 Following condition(s) must be satisfied simultaneously. (i) Discriminant 0 ; (ii) Sum of roots < 0; (iii) Product of roots 0 Now, D > 0 aligned & 4(a-3)^2 4 & (a-3)^2-1 0 & ( a -2)( a -4) 0 aligned a (- , 2) (4, )...(i) aligned & Also -2(a-3) 0 & a-3 0 & a 3...(ii) aligned And product of root(s) =1 0 a R array ll & ( i ) ( ii ) ( iii ) &