Class 12 > L21 : First Derivative Test - Application of Derivatives, Maths, Class 12

1. What is the first derivative test in calculus? |

Ans. The first derivative test is a method used in calculus to determine the intervals of increase or decrease of a function. It helps in identifying the local maxima and minima points of a function by analyzing the sign changes of its first derivative.

2. How is the first derivative test applied to find local extrema? |

Ans. To apply the first derivative test, follow these steps:
1. Find the critical points of the function by solving f'(x) = 0 or f'(x) does not exist.
2. Use the critical points to divide the number line into intervals.
3. Test a point from each interval in the first derivative to determine its sign.
4. If the sign changes from positive to negative, there is a local maximum at that point. If the sign changes from negative to positive, there is a local minimum.

3. Can the first derivative test be used to find global extrema? |

Ans. No, the first derivative test can only be used to find local extrema. To find global extrema, one needs to consider the endpoints of the function's domain and compare the values at those points.

4. What is the importance of the first derivative test in real-life applications? |

Ans. The first derivative test plays a crucial role in optimization problems. It helps in finding the maximum or minimum values of a function, which are often of interest in various real-life scenarios, such as maximizing profit, minimizing cost, optimizing resource allocation, and determining the optimal value for a given situation.

5. Are there any limitations or assumptions associated with the first derivative test? |

Ans. Yes, there are certain limitations and assumptions associated with the first derivative test. Some of them include:
- The function should be differentiable in the given interval.
- The critical points should be finite and isolated.
- The function should be continuous in the given interval.
- The test may not provide conclusive results if the function has points of inflection or vertical tangents.

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