Question 46 of 82beginner🔧 ApplyShort Answer2 marks

What is the maximum value of the function sin x + cos x?

Correct Answer

To find the maximum value of f(x) = sin x + cos x, we use the derivative method. Taking the derivative, we get f'(x) = cos x - sin x. Setting f'(x) = 0 for critical points, we have cos x = sin x, which gives tan x = 1. Therefore, x = π/4 + nπ.

At x = π/4, we evaluate: f(π/4) = sin(π/4) + cos(π/4) = 1/2\sqrt{2} + 1/2\sqrt{2} = 2\sqrt{2}.

Thus, the maximum value of the function sin x + cos x is 2\sqrt{2}.

Exercise: EXERCISE 6.3 | Q: 9 | (Page 29)
For More Understanding

Explanation

The question asks for the maximum value of sin x + cos x. Using the derivative method shown in the context (Section 6.4), we find critical points by setting the derivative to zero. The derivative f'(x) = cos x - sin x equals zero when tan x = 1, i.e., at x = π/4 + nπ. Evaluating at x = π/4 gives the maximum value 2\sqrt{2}.

Solution Steps

  1. Step 1: Find the derivative f'(x) = cos x - sin x

  2. Step 2: Set f'(x) = 0 to get cos x = sin x, or tan x = 1

  3. Step 3: Critical points are at x = π/4 + nπ

  4. Step 4: Evaluate f(π/4) = sin(π/4) + cos(π/4) = 1/2\sqrt{2} + 1/2\sqrt{2} = 2\sqrt{2}

  5. Step 5: The maximum value is 2\sqrt{2}

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