Let f(x)=|sinx|+|cosx|.
Domain : 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 the range of the function f(x) is [-√2,√2].
thankyou.
Question : If $x\sin^{3}\theta +y\cos^{3}\theta=\sin\theta\cos\theta$ and $x\sin\theta-y\cos\theta=0$, then the value of $\left ( x^{2}+y^{2} \right )$ equals:
Option 1: $1$
Option 2: $\frac{1}{2}$
Option 3: $\frac{3}2$
Option 4: $2$
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