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

We are given that the lines \( L_1 \) and \( L_2 \) intersect the curve \( y = f(x) \) at three and four distinct points, respectively. We want to determine the behavior of the derivatives of \( f(x) \). - First Derivative \( \frac{df}{dx} \): The fact that \( L_1 \) intersects \( y = f(x) \) at three distinct points and \( L_2 \) intersects \( y = f(x) \) at four distinct points suggests that \( f(x) \) changes slope at each intersection. The first derivative \( \frac{df}{dx} \) corresponds to the slope of the curve. Since the curve has three points of intersection with \( L_1 \), \( f(x) \) changes direction at least three times, which means \( \frac{df}{dx} \) will attain various values. This makes option (A) not true. - Second Derivative \( \frac{d^2f}{dx^2} \): The second derivative describes the concavity of the curve. It is not directly tied to the number of intersections but rather to how the curve bends. While it might vanish at some points, there is no immediate guarantee that it will vanish exactly once, so (B) is not necessarily true. - Third Derivative \( \frac{d^3f}{dx^3} \): The third derivative represents the rate of change of the concavity. For the curve to intersect the lines \( L_1 \) and \( L_2 \) at multiple distinct points, there must be points where the curve changes concavity, implying that \( \frac{d^3f}{dx^3} \) must vanish at least once. This matches option (C), so this is the correct answer. - Fourth Derivative \( \frac{df}{dx} \) at a specific value: The first derivative is not restricted from attaining specific values like \( \frac{7}{2} \). Hence, option (D) is not guaranteed. Thus, the correct answer is (C): \( \frac{d^3f}{dx^3} \) vanishes at least once on \( \mathbb{R} \).