Question

Find the range of the following function y=sinx×cosx

Answer 1

Domain of the function f(x) will be the intersection of domains of sinx and cosx. As the domain of sinx as well as cosx is (-∞,∞), thus the domain of the funtion f(x) will the the intersection of the two domains which comes out to be (-∞,∞) that is, that x can take any real value ranging from -∞ to +∞.

Therefore, the domain is (-∞,+∞).

Range : Range of any continuous funtion lies inbetween the minimum and maximum value of that function.

For finding the minimum and maximum of the function f(x), differentiate f(x) w.r.t. x and equate it with 'zero'.

Mathematically, df(x)/dx=0

So, d(sinx+cosx)/dx=0

i.e. cosx-sinx=0

=> cosx=sinx

=> x=π/4, 5π/4, 9π/4 and so on.

Taking x=5π/4 for f(x) to be minimum, f(x)=-2/√2=-√2.

Taking x=π/4 for f(x) to be maximum, f(x)=2/√2=√2.

Thus