- How to find the maximum and minimum points
- How to determine the speed of movement
- How to decide the limits
Find the value of the variable of this expression. These will be the values at which this derivative will be 0. To do this, substitute arbitrary digits in the expression instead of x, for which the whole expression will become zero. For example:
2-2x2 = 0
(1-x) (1 + x) = 0
x1 = 1, x2 = -1
Values of the function and the maximum and minimum points
The highest value of the function
Smallest function value
As the godfather said: "Nothing personal." Only derivatives!
Article How to calculate derivatives? I hope you learned, without this it will be problematic further.
12 the task on statistics is considered difficult enough, and all because the guys did not read this article (joke). In most cases, carelessness is to blame.
12 task can be of two types:
- Find the maximum / minimum point (ask to find the value of "x").
- Find the largest / smallest value of the function (asked to find the values of "y").
Find the high / low point
- Take the derivative of the proposed function.
- Set it to zero.
- The “x” found or found will be the minimum or maximum points.
- Determine the signs using the interval method and select which point is needed in the task.
Find the maximum point of the function
- Equate it to zero:
- We got one x value, to find the signs we substitute −20 to the left of the root and 0 to the right of the root in the transformed derivative (the last line with the transformation):
That's right, first the function increases, then decreases - this is the maximum point!
Answer: −15
Find the minimum point of the function
- Transform and take the derivative:
- It turns out one root “−2”, but do not forget about “−3”, it will also affect the sign change.
- Fine! First the function decreases, then it grows - this is the minimum point!
Find the largest / smallest function value
- Take the derivative of the proposed function.
- Set it to zero.
- The found "x" will be the minimum or maximum point.
- Determine the signs using the interval method and select which point is needed in the task.
- In such tasks, a gap is always specified: the Xs found in paragraph 3 must be included in this gap.
- Substituting the obtained maximum or minimum point in the initial equation, we obtain the largest or smallest value of the function.
Find the largest value of the function on the interval [−4, −1]
- Transform and take the derivative:
- “3” does not go into the interval [−4, −1]. So, it remains to check “−3” - is this the maximum point?
- It fits, first the function increases, then decreases - this is the maximum point, and in it will be the largest value of the function. It remains only to substitute the original function:
Find the largest value of the function on the segment [0, 1,5π]
- Take the derivative:
- We find what sin (x) equals:
- But this is impossible! Sin (x).
- It turns out that the equation has no solution, and in such situations it is necessary to substitute the extreme values of the gap in the original equation:
- The largest value of the function is “11” at the maximum point (on this segment) “0”.
- 70% of the errors are that the guys do not remember that in response to the largest / smallest value of the function, you need to write "y" and write "x" to the maximum / minimum point.
- Does the derivative have no solution when finding function values? Never mind, substitute the extreme points of the gap!
- The answer can always be written as a number or decimal. No? Then redefine the example.
- In most tasks, one point will be obtained and our laziness to check the maximum or minimum will be justified. We got one point - you can safely write in response.
- But with the search for the value of the function, you should not do this! Check that this is the desired point, otherwise the extreme values of the gap may turn out to be more or less.
