Paragraph: Let f(x) be a polynomial of degree 3 such that f(0)=1, f(1)=2 and zero is a critical point of f(x) having no local extreme. Question: If the value of definite integral _ -1 ^1 f(x) x^2+7 d x is equal to 2 ( a +1 b +c ) then the value of (a+b+c) , is:

Step-by-step solution

array lll f^ (x)=3 a x^2 & & f(x)=a x^3+b f(0)=b=1 & and & f(1)=a+1=2 a=1 array Hence, f(x)=x^3+1 (i) Limit =e^ _ x 0 x^3 x-x =e^3 (ii) aligned & D.I. = _ -1 ^1 x^3+1 x^2+7 d x=2 ₀^1 1 x^2+7 d x=2 ( (x+ x^2+7 ) )₀^1=2 ( 8 +1 7 ) & a+b+c=8+7+0=15 aligned