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# RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima (Ex 18.1) Exercise 18.1

Last updated date: 02nd Aug 2024
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## RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima (Ex 18.1) Exercise 18.1 - Free PDF

Free PDF download of RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima Exercise 18.1 solved by Expert Mathematics Teachers on Vedantu.com. All Chapter 18 - Maxima and Minima Ex 18.1 Questions with Solutions for RD Sharma Class 12 Maths to help you to revise the complete Syllabus and Score More marks.

Competitive Exams after 12th Science

## Maxima and Minima

The notion of derivatives is used to find maxima and minima in calculus. We locate the locations where the gradient is zero and these points are called turning points/stationary points, as we know the idea of derivatives offers us information about the gradient/ slope of the function. These are the points that correspond to the function's largest and smallest values (locally).

Maxima/minimum knowledge is vital for our day-to-day applicable challenges. The method of determining the absolute maximum and minimum is also discussed in the article.

This page contains solutions to maxima and minima calculus problems.

### Local Maxima and Local Minima

A local maximum point on the graph of a function is a point (x,y) on the graph whose y coordinate is bigger than all other y coordinates on the graph at nearby points (x,y).

In other words, if there is an interval (a,b) with an x b and f(x) f(z) for every z in the set, then (x,f (x)) is a local maximum (a,b). Similarly, if (x,y) has the smallest y coordinate locally, it will be determined as the local minimum point.

To put it another way, (x,f(x)) is a local minimum if there is an interval (a,b) with an x b and f(x) f(z) for every z in the set (a,b). A local minimum or maximum is referred to as a local extreme.

## FAQs on RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima (Ex 18.1) Exercise 18.1

1. What are the various applications for maxima and minima?

The following are some examples of maxima and minima applications:

For an engineer, a function's maximum and minimum values can be utilized to determine its boundaries in real life. For example, if you can determine an adequate function for a car's speed, determining the maximum possible speed of a train can help you select the equipment that will be sufficient enough to withstand the pressure caused by such high speeds, and it will be helpful for them to design the brakes and wheels, among other things, to ensure that the car runs smoothly.

For an economist, the maximum and lowest values of the total profit function aid in determining the boundaries that the corporation should place on employee pay to avoid losses.

For a doctor, the maximum and lowest values of the function define the total blood level in the bloodstream, allowing the doctor to decide the dosage that should be prescribed to various individuals to restore normal blood pressure levels. To learn more, click here.

2. What are the absolute maxima and absolute minima of a function?

If (a,b) is a point in the xy-area plane's D, Z = f(x,y) is said to have

When f(x,y) f(p,q) holds for all (x,y) in D, there is an absolute Maxima in area D at (p,q). When f(x,y) f(p,q) holds for all (x,y) in D, there is an absolute Maxima in area D at (p,q). Global maxima and minima are other names for absolute maxima and minima.

When f(x,y) is continuous on D, absolute maxima and minima always appear someplace on D for a single variable function. The Absolute Maxima and Absolute Minima are found by following these steps:

Locate the essential interior spots. It entails computing f(x,y) and then set it equal to 0. On the boundary, maximize or reduce f(x,y). Compare the f(x,y) values you got from the first two stages above.

3. What are stationary points vs turning points?

The spots on a graph where the slope reaches zero are known as stationary points. In other words, the function's tangent becomes horizontal, with dy/dx = 0. Figures A,B, and C below illustrate all of the stationary spots. Points A and B are turning points because the curve alters its direction. If it was traveling uphill, it would go downward, and vice versa. However, even though the graph is flat for a brief time, it continues to travel down from left to right, point C is not a turning point.

4. What is the first-order derivative test?

Assume that f is a function defined in the open interval I. And let f be continuous in I at critical point c, with f'(c) = 0.

1. C is the point of local maximum if f'(x) changes sign from positive to negative as x increases through point c. And the maximum value is f(c).

2. If the sign of f'(x) changes from negative to positive as x increases through point c, c is the local minima point. And f(c) is the smallest value.

3. If the sign of f'(x) does not change as x increases through c, then c is neither a local nor a local maxima point. It will be known as the inflection point.

5. What are the properties of maxima and minima function?

1. If f(x) is a continuous function in its domain, there should be at least one maximum and one minimum between equal values of f(x) (x).

2. The maxima and minima alternate. In other words, there is one maxima between two minima and vice versa.

3. If f(x) goes to infinity as x approaches an or b, and f'(x) = 0 only for one value x, i.e. c between a and b, then f(c) is the smallest and least value. If f(x) goes to – as x tends to an or b, then the maximum and highest value is f(c). To get free study materials, access the Vedantu app and website.