Sol. We know that two curves touch each other at a point of intersection iff their slopes at the point are equal.

First we find the points of intersection of y = x 2 sin 3x and y = x 2 . At the points of intersection, we have

x 2 sin 3x = x 2 ⇒ sin 3x = 1

⇒ 3x = 2n π +

⇒ x = + , n ∈ Z

Now,

y = x

2 sin 3x

⇒ = 2x sin 3x + 3x 2 cos 3x

⇒ = 2 = + ... (i)

and,

y = x 2 = 2x

⇒ = 2 = + ... (ii)

From (i) and (ii), we find that the curves y = x

2 sin 3x and y = x 2 have the same slope at their points of intersection. So, they touch each other.

Let us now find the points of intersection of y = x 2 sin 3x and y = –x 2 . At the points of intersection, we have

x 2 sin 3x = –x 2 ⇒ sin 3x = –1

⇒ 3x = 2n π – , n ∈ Z

⇒ x = – , n ∈ Z

Now,

y = x

2 sin 3x

⇒ = – 2 = – + ... (iii)

and,

y = –x 2 ⇒ = – 2 = + ... (iv)

From (iii), (iv), we observe that the curves y = x

2 sin 3x and y = –x 2 have the same slopes at their points of intersection. Hence, they touch each other.

It is obvious that the sets of points given by x = + and x = – , n ∈ Z occur alternately.

Hence, y = x 2 sin 3x touches y = x 2 and y = –x 2 alternately at infinite number of points.