For all real values of x, the expression 5 sin(x) + 12 cos(x) + r is required to be greater than or equal to 0. What is the smallest value of r that satisfies this condition?

Introduction:
This question tests the minimum/maximum value of a trigonometric linear combination of sin(x) and cos(x). Expressions of the form A sin(x) + B cos(x) have a fixed amplitude equal to sqrt(A^2 + B^2). That means the expression varies between -sqrt(A^2 + B^2) and +sqrt(A^2 + B^2) as x changes. To ensure 5 sin(x) + 12 cos(x) + r is always at least 0, r must shift the entire range upward so that the minimum becomes 0. Finding the smallest such shift gives the required minimum r.

Given Data / Assumptions:

• Expression: 5 sin(x) + 12 cos(x) + r
• Must satisfy: 5 sin(x) + 12 cos(x) + r ≥ 0 for all real x
• Amplitude rule: A sin(x) + B cos(x) ranges from -sqrt(A^2 + B^2) to +sqrt(A^2 + B^2)

Concept / Approach:
Compute the minimum of 5 sin(x) + 12 cos(x). It is -sqrt(5^2 + 12^2). Then require: minimum + r ≥ 0, so r ≥ sqrt(5^2 + 12^2). Choose the smallest r that satisfies this, which is exactly that amplitude value.

Step-by-Step Solution:
A = 5 and B = 12 Amplitude = sqrt(A^2 + B^2) = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13 Minimum of 5 sin(x) + 12 cos(x) = -13 We need -13 + r ≥ 0 for all x So r ≥ 13 Smallest such value is r = 13

Verification / Alternative check:
If r = 13, the expression becomes 5 sin(x) + 12 cos(x) + 13. Since 5 sin(x) + 12 cos(x) is at least -13, the total is at least 0. If r were even slightly less than 13, then at the x where 5 sin(x) + 12 cos(x) reaches -13, the expression would become negative, violating the requirement.

Why Other Options Are Wrong:
-13 and -11 shift the expression downward, making it negative for many x values. 0 and 11 are not enough to offset the minimum -13, so the expression would still go below 0. Only 13 ensures the minimum reaches exactly 0 without being negative.

Common Pitfalls:
Assuming sin and cos can both be 1 at the same time, adding coefficients as 5+12, or forgetting to take the square root when computing amplitude.

Final Answer:
The smallest value of r is 13.