The value of b for which the function f(x) = x + cos x + b is strictly decreasing over R is
In mathematics, a function is a relation that uniquely associates elements from one set (domain) to another set (codomain). Each input corresponds to exactly one output, ensuring well-defined mapping. Functions model relationships, transformations and dependencies in various fields like calculus, algebra and geometry, enabling analysis of patterns and behavior.
Class 12 Maths Chapter 6 focuses on Applications of Derivatives for CBSE Exam 2024-25. It covers rate of change of quantities increasing and decreasing functions tangents and normals maxima and minima. These concepts are applied to solve real-life problems in physics economics and engineering while enhancing problem-solving skills and logical thinking.
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In order to determine the value of b for which the function f(x) = x + cos x + b is strictly decreasing over ℝ, we start by analyzing the derivative of the function. Let us differentiate f(x):
f'(x) = d/dx(x + cos x + b) = 1 – sin x
Hence for being strictly decreasing, the derivative must be negative for all values of x; i.e., f'(x) < 0.
1 – sin x 1
But the function sin x can never be more than 1 for any real number x. Thus, b cannot be taken such that for all real number x, f'<0.
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