Judge the monotonicity, boundedness of the function?

Whether a constant multiplied by a bounded function is a bounded function.

x-squared is an unbounded function.

Multiplying an unbounded function by a bounded function is not a bounded function. So d wrong.

x 2 is an even function, cosx is also an even function, and the product of two even functions is still an even function. So choose A. As for boundedness, it is clear that when x tends to infinity, cosx is bounded, but x 2 is unbounded, and the unbounded multiplies the bounded, and the result is still unbounded.

y=x 2cosx, which is an even function.

To determine the parity of functions, there are definition methods, image methods, and arithmetic function parity methods.

This problem can be judged by the arithmetic function parity method, it couldn't be simpler, x 2 and cosx are even functions, and even functions Even functions are still even functions.

The law of parity of arithmetic functions is as follows:

odd function + - odd function = odd function; even function + - even function = even function; odd function + - even function = non-odd and non-even function;

Odd Functions Odd Functions = Even Functions; even function + - even function = even function; Odd Functions Even = Odd Functions.

This student, this question is very easy to solve, y=x cosx is an even function, f(-x)=(-x) cos(-x)=x cosx=f(x), and the derivative can be obtained by finding f'(x)=2xcosx-x sinx, it is found that it is not a monotonic increasing function, and its monotonicity.

It's always changing, so it's not a bounded function!

f(x)=x²cosx

The domain is defined as rf( x) = ( x ) cos( x) = x cosx=f(x).

So f(x) is an even function.

x is an unbounded function, and x cosx is also an unbounded function.

Because the square of x is not bounded, the whole function is unbounded.

Addition, subtraction, multiplication, and division of a bounded function is still bounded, which is ......... who said

Obviously, when x tends to positive infinity, x2 is positive infinity, multiplied by cosx (can take the value -1 to 1), the boundary can be found?

The judgment methods include the derivative method, the definition method, the property method, and the methodComposite functionsSame increase and different subtraction.

1. Derivative method: first find the derivative of the function, let itDerivativesEqual to zero, the value of x is obtained, and the relationship between x and the derivative function is judged, when the derivative function is greater than zeroIncrement function, less than zero isSubtract the function

2. Definition method: Let x1 and x2 be any two numbers on the domain of the function f(x) definition, and x1 x2, if f(x1) f(x2), then this function is an increasing function; On the contrary, if f(x1) f(x2), then this function is a subtraction function.

3. Properties method: if the functions f(x) and g(x) have monotonicity in the interval b.

then in interval B there is:

f(x) has the same monotonicity as f(x) c (c is a constant).

f(x) is the same monotonicity as c f(x) when c 0 has the same monotonicity and when c 0 has the opposite monotonicity.

When f(x) and g(x) are both increase (decrease) functions, then f(x) and g(x) are both increase and decrease functions.

Denote. The first thing to understand is that a function is a correspondence that occurs between sets. Then, it is necessary to understand that there is more than one function relationship between a and b. Finally, it is important to understand the three elements of a function.

The correspondence of functions.

It is usually expressed analytically, but a large number of functional relationships cannot be expressed analytically, and can be expressed in images, **, and other forms.

Conception. In a process of change, the amount of change is called a variable (in mathematics, it is often x, and y changes with the change of x value), and some values do not change with the variable, we call them constants.

Argument. Function): A variable associated with a quantity in which any value can find a fixed value.

1. Boundedness.

It is the boundary on the y-axis, such as y=sinx, -1<=y<=1, this is the boundedness of the equation, and the boundedness is artificially widened, and the range of values of x can be limited, such as y=tanx, where x [-1,1] is bounded.

The following methods are commonly used to determine the boundedness of a function.

1. A continuous function on a closed interval must be a bounded function.

2. Appropriately enlarge or shrink the relevant expressions to derive their boundaries.

3.Image judgment using basic elementary functions.

2. Monotonicity.

Monotony increases <>

Monotony reduces <>

3. Parity.

Parity is premised on the fact that the domain is defined as symmetrical with respect to the origin.

Odd function images are symmetric with respect to the origin, while even functions are symmetric with respect to the y-axis.

Fourth, periodicity.

Let the period of the function f(x) be t, then the period of f(ax+b) is. The sufficient and necessary conditions for f(x) with respect to the symmetry of the straight line x=t are: f(x)=f(2t-x).

1.Judging the boundedness of the Qingbian function: the image of the function can be analyzed, and if the image of the function increases or decreases monotonically in a certain interval, then the function is bounded in that interval; In addition, it is also possible to use the mathematical method of separation, if the function is buried in the upper and lower bounds in a certain interval, then the function is bounded in that interval.

2.Judging the monotonicity of the function: the image of the function can be analyzed, if the image of the function increases or decreases monotonically in a certain interval, the function is monotonionic in that interval; In addition, mathematical analysis can also be used, if the derivative of a function satisfying a function in a certain interval is greater than zero or less than zero, the function is monotonic in that interval.

Draw an image by yourself, the corresponding interval of the upward trend is a monotonic increasing interval, and the downward trend is a decreasing interval, and you can add 2k to the upper and lower bounds of the corresponding interval respectively (k belongs to an integer by default), so that you don't have to memorize cumbersome mathematical formulas. As shown in Fig

In middle school, only these three are required, so I wish you a happy study.

There are two main ways to judge the monotonicity of a function:

The first method is the definition method, which is also a judgment method proposed by high school mathematics, which is mainly used for relatively simple functions or composite functions.

The second method, the derivative method, judges the monotonicity of the function by finding the derivative of the function and judging the positive and negative of the derivative function. This method can handle both simple and complex functions.

Definition: <>

In the derivative method, in the specified area, the first-order finch slow guide is greater than 0, and the single side adjustment increases, and the number of modulus decreases on the contrary.

The monotonicity of a function can also be called the addition or decrease of a function.

Methods: 1. Image observation method.

As mentioned above, on the monotonic interval, the image of the increasing function is upward, and the image of the decreasing function is decreasing. Therefore, in a certain interval, the function corresponding to the function image that has been rising increases monotonically in that interval; The function image that has been decreasing corresponds to a monotonically decreasing function in that interval.

2. Derivative method.

Derivatives are closely related to the monotonicity of functions. It is another way to study functions, opening up many new avenues for it. Especially for specific functions, the use of derivatives to solve the single-pin tonality of the function is clear, the steps are clear, fast and easy to master, and the use of derivatives to solve the monotonicity of the function requires proficiency in the basic derivative formula.

If the function y=f(x) is derivable (differentiable) in the interval d, if there is always f at x d'(x)>0, then the function changes to auspicious cherry y=f(x) monotonically increases in the interval d; Conversely, if x d, f'If the kernel plexus (x) < 0, the function y=f(x) is said to decrease monotonically in the interval d.

1.Judge the monotonicity of a function.

The monotonicity of derivative functions is closely related to the notation of derivatives. In turn, we can judge the monotony of a function by the sign of the derivative.

Let the function f(x) be continuous on [a,b] and be derivable in (a,b), then there is.

1) If in (a, b) f'(x) >0, then the squire function f(x) increases monotonically in (a,b);

2) If in (a, b) f'(x) <0, then the function f(x) decreases monotonically in (a,b).

According to the theorem a general step can be derived to discuss the monotonicity of a function:

1) determine the domain of the function f(x);

2) find the points with f(x)=0 and the points that f(x) does not exist, and divide the definition domain into several sub-intervals with these points as the demarcation points.

3) The symbols of f(x) in each interval are discussed separately to determine the monotonicity of the function.

If f'(x0)=0, then x0 is said to be the station point of the function f(x).

Method: Derivation, stationing, division of definition domain, judgment. Examples:

Function increase or decrease judgment formula:

Same increase and different subtraction. Increase + increase = increase.

minus + minus = minus.

Increase-decrease = increase.

Decrease - increase = decrease. With traces.

How to determine the increase or decrease of a function:

1.Basic Function Method.

The method of judging the monotonicity of a function by using the monotonicity of familiar basic functions (primary, quadratic, inverse proportional, exponential, logarithmic, trigonometric and other functions) is called the basic function method.

2.Imagery.

The method of judging the monotonicity of a function by using the image of the function is called the image method. The image gradually rises from left to right<=> is an increasing function. The method of judging the monotonicity of a function by using the function image from the left is called the image method.

The image gradually rises from left to right<=> is an increasing function. The image gradually descends from left to right<=> is a subtractive function.

3.Definition: <>

The method of judging the monotonicity of a function by using the definition of monotonicity is called the definition method. Let x1, x2 d, x1) <=x) be the increasing (subtracting) function on d. The process is to take the value one by one, make the difference, deform one by one, judge the symbol one by one, and conclude it.

In fact, this is also the process of proving monotonicity.

4.Function operation algorithms.

The method of judging the monotonicity of a function by using the sum and difference product quotient of the monotonic function obtained by the four operations is called the function operation algorithm. Let f and g be the increasing functions, then on the monotonic increase interval of f, or on the intersection of the monotonic increase interval of f and g, there is the following conclusion:

f+g is an increment function.

f is a subtraction function.

1 f is a subtraction function (f>0).