Question: All the points on the curve y = $\sqrt{x + \sin x}$ at which the tangent is parallel to x-axis lie...

Question

All the points on the curve y = $\sqrt{x + \sin x}$ at which the tangent is parallel to x-axis lie on a/an

Answer 1

y = $\sqrt{x + \sin x}$

Let at P(x₁, y₁) tangent be parallel to x-axis. So

$\left( \frac{dy}{dx} \right)_{(x_{1},y_{1})}$ = 0

Ž $\frac{1 + \cos x_{1}}{2\sqrt{x_{1} + \sin x_{1}}}$ = 0 Ž 1 + cos x₁ = 0

Ž cos x₁ = –1 Ž x₁ = p Ž sin x₁ = 0 ....(i)

Now (x₁, y₁) lies on given curve

\ y₁² = x₁ + sin x₁

Ž y₁² = x₁ (sin x₁ = 0 from (i))

\ locus of (x₁, y₁) is y² = x

which is parabola.

Question: All the points on the curve y = \(\sqrt{x + \sin x}\) at which the tangent is parallel to x-axis lie...