On the curve x3=12y, the abscissa changes at a faster rate than the ordinate. Then, x belongs to the interval

Question

On the curve x3=12y, the abscissa changes at a faster rate than the ordinate. Then, x belongs to the interval (A)  (-2,2)  (B)  (-1,1)  (C) (0, 2) (D) None of these

Tardigrade Admin · Accepted Answer

Given, x3=12y ⇒ 3x2(dx/dy)=12 ⇒ (dx/dy)=(4/x2) But it is given | (dy/dx) |>1 ∴ (4/x2)>1 ⇒ x2-4<0 ⇒ -2

Q. On the curve $x^{3} = 12 y,$ the abscissa changes at a faster rate than the ordinate. Then, $x$ belongs to the interval

$(- 2, 2)$

$(- 1, 1)$

(0, 2)

None of these

Given, $x^{3} = 12 y$ $\Rightarrow$ $3 x^{2} \frac{d x}{d y} = 12$ $\Rightarrow$ $\frac{d x}{d y} = \frac{4}{x ^{2}}$ But it is given $∣ ∣ \frac{d y}{d x} ∣ ∣ > 1$ $∴$ $\frac{4}{x ^{2}} > 1$ $\Rightarrow$ $x^{2} - 4 < 0$ $\Rightarrow$ $- 2 < x < 2$

Q. On the curve x3=12y, the abscissa changes at a faster rate than the ordinate. Then, x belongs to the interval

(−2,2)

(−1,1)