Let be a binary operation on Q defined by
Show that is commutative as well as associative. Also find its identity element, if it exists.
Find the equations of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
Find the values of x for which f(x) = [x(x - 2)]2 is an increasing function. Also, find the points on the curve where the tangent is parallel to x-axis.
So, the tangents to curve f ( x ) is parallel to the x-axis if x = 0, x = 1 or x = 2.
Now points 0, 1 and 2 will divide the number line into 4 disjoint intervals
Which gives us x = 0, 1, 2
Hence, x = 0, y = 0
x = 1, y = 0
x = 2, y = 0
Required points are ( 0, 0 ), ( 1, 1 ), ( 2, 0 ).
Show that the right circular cylinder, open at the top, and of given surface area and maximum volume is such that its height is equal to the radius of the base.