Question

Find the maximum and minimum value of sin θ + cos θ

Answer 1

Let, y=sinx+cosx

Now differentiating both sides with respect to x, we get:

dy/dx=cosx-sinx

To find maximum or minimum value (dy/dx) must be equal to zero.

Hence,  cosx-sinx=0

              cosx=sinx

Above equation satisfies when, x=π/4 and π+π/4

Now we have to find when maxima occur and when minima. So we have to double differentiate y.Let double differentiation of y is z.

Z=-sinx-cosx

For x=π/4,     Z=-√2. Z comes negative. So at x=π/4 maxima occurs.

And fotx=π+π/4,     z=√2. Z comes positive. So at x=π+π/4 minima occurs.

So, maximum value of sinx +cosx=√2 and minimum value of sinx +cosx=-√2.

I hope the above solution is clear to you.

Answer 2

Maximum value = √(1²+1²) = √2

Minimum value = -√(1²+1²) = -√2