Maximum value of (sinx + cosx) is
(a) 1
(b) 2
(c)\[\sqrt{2}\]
(d) \[\dfrac{1}{\sqrt{2}}\]

Hint: For maximizing the function f(x)=sinx + cosx, we will first multiply and divide the equation with \[\sqrt{2}\] that is \[\dfrac{\sqrt{2}(\sin x+\cos x)}{\sqrt{2}}\] then , as we know that \[\dfrac{1}{\sqrt{2}}\] can be written as \[\sin \dfrac{\pi }{4}\ and\ \cos \dfrac{\pi }{4}\] , therefore, the expression can be written as \[\sqrt{2}\left( \cos \dfrac{\pi }{4}\sin x+\sin \dfrac{\pi }{4}\cos x \right)\] then we can apply the formula for the sum of angles in the sin function. Also another important formula that would be used in the solution would be as follows \[sin\left( x+y \right)=sinx\cdot cosy+siny\cdot cosx\]

Complete step-by-step answer:
Now, in this question, we will first multiply and divide the function with \[\sqrt{2}\] . Then, we will write the value of \[\dfrac{1}{\sqrt{2}}\] as \[\sin \dfrac{\pi }{4}\ and\ \cos \dfrac{\pi }{4}\] .
Now, we will use the following formula which is \[sin\left( x+y \right)=sinx\cdot cosy+siny\cdot cosx\] to get a single sine function whose value, we can then maximize.

As mentioned in the question, we need to maximize the given expression and for that we would follow the exact procedure which is mentioned in the hint that is as follows
 We will first multiply and divide the equation with \[\sqrt{2}\] .
\[=\dfrac{\sqrt{2}(\sin x+\cos x)}{\sqrt{2}}\]
Then, as we know that \[\dfrac{1}{\sqrt{2}}\] can be written as \[\sin \dfrac{\pi }{4}\ and\ \cos \dfrac{\pi }{4}\] , therefore, the expression can be written as
 \[=\sqrt{2}\left( \cos \dfrac{\pi }{4}\sin x+\sin \dfrac{\pi }{4}\cos x \right)\]
Now, using the formula mentioned in the hint, we get
f(x)
\[\begin{align}
  & =\sqrt{2}\left( \sin x\cdot \cos \dfrac{\pi }{4}+\sin x\cdot \cos \dfrac{\pi }{4} \right) \\
 & =\sqrt{2}\left( \sin \left( x+\dfrac{\pi }{4} \right) \right) \\
\end{align}\]
Now, we know that over the entire domain of sin function, the maximum value that it can attain is 1, hence, the maximum value of the above mentioned expression is as follows
\[\begin{align}
  & =\sqrt{2}\left( \sin \left( x+\dfrac{\pi }{4} \right) \right) \\
 & =\sqrt{2} \\
\end{align}\]
so, the maximum value of the ( sinx + cosx ) is equal to \[\sqrt{2}\].
Hence, option ( c ) is correct.

So, the correct answer is “Option (c)”.

Note: The students can make an error in evaluating the maximum value of the given expression if they don’t have any idea about the procedure that is mentioned in the hint and also the students can get confused in finding the maximum value of the sin function at the end of the simplification.
Another method of solving this question is by taking derivative of the function that is
f(x)=sinx + cosx and then equating it to zero which will give the value of x at which the function will attain the maximum value.