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Question: The maximum value of \(\sin \theta + \cos \theta \) is: 1) 1 2) 2 3) 3 4) \(\sqrt 2 \)...

The maximum value of sin⁡θ+cos⁡θ\sin \theta + \cos \theta is:

  1. 1
  2. 2
  3. 3
  4. 2\sqrt 2
Explanation

Solution

The above given function can be stated as maximum or minimum value by finding its second derivative and substituting the value of theta which we will calculate after the first derivative, in the second derivative in order to find the sign convention of the second derivative if the value comes out to be negative then the function has maximum value else if the function has positive value then the function has minimum value.

Complete step-by-step solution:
Let's discuss the method of finding the maximum value in more details and then we will perform the calculations.
First we will find the derivative of the given function;
df(x)dx=0\dfrac{{df(x)}}{{dx}} = 0 and will equate it to zero in order to find the value of the unknown quantity be it x or angle theta in case of the given question.
After calculating the value of theta we will check whether the given function is minimum or maximum and second derivative is performed d2f(x)dx2\dfrac{{{d^2}f(x)}}{{d{x^2}}} .
If the value of the second derivative comes out to be negative then the given function has maximum value and if the value of the second derivative comes out to be positive then the function has minimum value.
Now, we will perform the calculations.
First derivative;
⇒d(sin⁡θ+cos⁡θ)dθ=0\Rightarrow \dfrac{{d(\sin \theta + \cos \theta )}}{{d\theta }} = 0 (derivate of cosθ\theta will be -sinθ\theta and derivative of sinθ\theta will be cosθ\theta )
⇒cos⁡θ−sin⁡θ=0\Rightarrow \cos \theta - \sin \theta = 0 (We will calculate the value of theta now)
⇒cos⁡θ=sin⁡θ\Rightarrow \cos \theta = \sin \theta
⇒tan⁡θ=1\Rightarrow \tan \theta = 1
⇒θ=tan⁡−11\Rightarrow \theta = {\tan ^{ - 1}}1
⇒θ=450orπ4\Rightarrow \theta = {45^0}or\dfrac{\pi }{4}
Now, we will perform the second derivative;
⇒d2(cos⁡θ−sin⁡θ)dθ2\Rightarrow \dfrac{{{d^2}(\cos \theta - \sin \theta )}}{{d{\theta ^2}}}
⇒−sin⁡θ−cos⁡θ\Rightarrow - \sin \theta - \cos \theta (Now we will substitute the value of theta in this second derivative)
⇒−sin⁡450−cos⁡450\Rightarrow - \sin {45^0} - \cos {45^0} (sin and cos 45 is equal to one upon root 2)
⇒−12−12\Rightarrow - \dfrac{1}{{\sqrt 2 }} - \dfrac{1}{{\sqrt 2 }} (On adding the two)
⇒−22\Rightarrow - \dfrac{2}{{\sqrt 2 }}
⇒−2\Rightarrow - \sqrt 2 (Negative sign indicates that the value of the function is maximum)

Option 4 is correct.

Note: Applications of maximum and minimum values can also be seen in our daily lives such as in order to find the maximum volume of the large containers, the design of the piping systems is often based on minimizing pressure drop which in turn minimizes required pump sizes and reduces cost.