the set of all the possible values of 'a' for which the function f(x)=5+(a-2)x+(a-1)x^2-x^3 has a local minimum value at - Maths - Application of Derivatives

Question

the set of all the possible values of 'a' for
which the function f(x)=5+(a-2)x+(a-1)x^2-x^3 has a local minimum
value at some x<1 and local maximum value at some x>1 is

Aman Choubey · Accepted Answer

Dear Student Differentiating it will give2(a-2)x - 3x2 In order for the function to have critical points in range x<1 2(a-2)x - 3x2 = 0⇒2(a-2) -3x = 0⇒2(a-2) =3x⇒2(a-2)3 =xFor x <12(a-2)3 < 1⇒a < 7/2For x > 12(a-2)3 > 1a > 7/2Regards

the set of all the possible values of 'a' for which the function f(x)=5+(a-2)x+(a-1)x^2-x^3 has a local minimum value at some x<1 and local maximum value at some x>1 is