If $27a+9b+3c+d=0$, then the equation $f(x)=4ax^3+3bx^2+2cx+d=0$ has at least one real root lying between $(0,3)$.

Let F(x) = ax^3 + bx^2 + cx + dx, meaning the given equation f(x) = 4ax^3 + 3bx^2 + 2cx + d is exactly the derivative F'(x). Evaluating the integral of F'(x) from 0 to 3 yields F(3) - F(0), which equals 27a + 9b + 3c + d, and since this equals 0, the area under F'(x) is zero. By Rolle's Theorem, if the accumulated integral is zero, the derivative f(x) must have at least one root between 0 and 3; thus, both the assertion and the reasoning correctly explain the property.

• Thanks
• Wrong explanation