For the curve \(4x^{2} + 9y^{2} = 72\), consider the following statements:
(A) At the point (3, 2), \(\displaystyle\frac{dy}{dx} = -\frac{2}{3}\).
(B) At the point (3, 2), \(\displaystyle\frac{d^{2}y}{dx^{2}} = -\frac{4}{9}\).
(C) At the point (−3, 2), \(\displaystyle\frac{dy}{dx} = -\frac{2}{3}\).
(D) At the point (3, −2), \(\displaystyle\frac{dy}{dx} = \frac{2}{3}\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (D) only
By implicit differentiation:
\( \displaystyle 8x + 18y\frac{dy}{dx} = 0 \)
\( \displaystyle \frac{dy}{dx} = -\frac{4x}{9y} \)
Differentiating again by the quotient rule:
\( \displaystyle \frac{d^{2}y}{dx^{2}} = -\frac{4}{9} \cdot \frac{y - x\frac{dy}{dx}}{y^{2}} \)
At (3, 2), \(\frac{dy}{dx} = -\frac{12}{18} = -\frac{2}{3}\), so
\( \displaystyle \frac{d^{2}y}{dx^{2}} = -\frac{4}{9} \cdot \frac{2 + 3 \times \frac{2}{3}}{4} = -\frac{4}{9} \)
So \(\frac{dy}{dx} = -\frac{4x}{9y}\), giving −2/3 at (3, 2), 2/3 at (−3, 2) and 2/3 at (3, −2), and \(\frac{d^{2}y}{dx^{2}} = -\frac{4}{9}\) at (3, 2).
Checking the statements
(A) At the point (3, 2), \(\frac{dy}{dx} = -\frac{2}{3}\). — Correct. \(-\frac{4 \times 3}{9 \times 2} = -\frac{2}{3}\).
(B) At the point (3, 2), \(\frac{d^{2}y}{dx^{2}} = -\frac{4}{9}\). — Correct. \(-\frac{4}{9} \cdot \frac{2 + 2}{4} = -\frac{4}{9}\).
(C) At the point (−3, 2), \(\frac{dy}{dx} = -\frac{2}{3}\). — Incorrect. \(-\frac{4 \times (-3)}{9 \times 2} = \frac{2}{3}\).
(D) At the point (3, −2), \(\frac{dy}{dx} = \frac{2}{3}\). — Correct. \(-\frac{4 \times 3}{9 \times (-2)} = \frac{2}{3}\).