Question

Let L₁ denote the line y = 3x + 2 and L₂ denote the line y = 4x + 3. Suppose that f: ℝ → ℝ is a four times continuously differentiable function such that the line L₁ intersects the curve y = f(x) at exactly three distinct points and the line L₂ intersects the curve y = f(x) at exactly four distinct points. Then, which one of the following is TRUE ?

• A. \(\frac{df}{dx}\) does not attain the value 3 on \(\R\)
• B. \(\frac{d^2f}{dx^2}\) vanishes at most once on \(\R\)
• C. \(\frac{d^3f}{dx^3}\) vanishes at least once on \(\R\)
• D. \(\frac{df}{dx}\) does not attain the value \(\frac{7}{2}\) on \(\R\)

Answer 1

The Correct Option is C

The correct option is (C) : \(\frac{d^3f}{dx^3}\) vanishes at least once on \(\R\).