A function is a rule that assigns each input exactly one output. It maps elements from a domain to a codomain. Functions are represented as f(x), where x is the input. Examples include linear, quadratic and trigonometric functions, such as f(x) = x².
Class 12 Maths Chapter 6 Applications of Derivatives is an important topic for the CBSE Exam 2024-25. It includes concepts like finding the rate of change of quantities increasing and decreasing functions maxima and minima tangents and normals and approximations. These concepts help in solving real-life problems related to physics economics and engineering.
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To find the interval for which the function y = x³ + 6x² + 6 is increasing, we will look at its derivative.
Compute the derivative of the function:
y'(x) = d/dx(x³ + 6x² + 6) = 3x² + 12x
Determine where the derivative is positive, so the function is increasing:
y'(x) > 0
3x² + 12x > 0
Factor the expression:
3x(x + 4) > 0
This inequality holds when x 0. Thus, the function is increasing in the intervals (-∞, -4) and (0, ∞).
Conclusion:
The correct intervals where the function is increasing are (-∞, 0) U (4, ∞).
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