Question

The equation 3x² + 4ax + b = 0 has at least one root in ] 0, 1[ if :

• A. 4a + b + 3 = 0
• B. 2a + b + 1 = 0
• C. b = 0, a = $- \frac{4}{3}$
• D. None of these

Answer 1

Answer: (B)

2a + b + 1 = 0

Answer 2

Consider the function f(x) = x³ + 2ax² + bx

obviously f(x) is continuous is [0, 1] and differentiable in ]0, 1[

Also f(0) = 0 if f(1) = 1 + 2a + b = 0 then all condition of Roll‘s theorem, are satisfied.

∴ f ′(3) = 0 for at least one c in ]0, 1[

Hence f ′(x) = 3x² + 4ax + b = 0 at least one root in ]0,1[ so 1 + 2a + b = 0