Follow finding the maximum value of the function, which in the interval has a finite number of critical points. To do this, calculate its value at all points and at the ends of the segment. From the resulting numbers, choose the highest. The method of finding the greatest value of the expression is used to solve various application tasks.
To do this, run the following steps: translate the task into the language of the function, select x, and through him to Express the required rate as a function of f(x). Using Analytics, find the largest and smallest values of the function at a certain interval.
Use the following examples to find the value of the function. Find the values of the function y=5-square root (4 – x2). Using the definition of the square root, we get 4 - x2 > 0. Solve the quadratic inequality, we get that -2 < x < 2. Divide the resulting interval into two, you get two inequalities -2 < x < 0 and 0 < x < 2.
Erect in the square each of the inequalities, then multiply all three parts by -1, add to them 4. Then enter the auxiliary variable and make the assumption that t = 4 - x2, where 0 < t < 4. A function of y, is the square root of the variable t on the interval is increasing and continuous. Therefore, the greatest value of the function happens at the end of the period.
Perform the inverse change of variables, as a result, you will get the following inequalities: 0 < root of (4 – x2) < 2. Add to all portions 5, before multiplying by -1, you get 3 < 5 - root of (4 – x2). < 5. Thus, the set of values of the function y = 5 - square root (4 – x2) is the interval [3; 5], and the largest value, respectively, 5.
Use the method of application properties of continuous functions to determine the highest value of the expression. In this case, use the numeric values taken by the expression at the specified interval. Among them, there is always a least value m and maximum value M. Between these numbers is a set of values of the function.
Advice 2: How to find multiple values
When we are dealing with functions, we have to find the domain of the function and the set of values of the function. This is an important part of the overall study algorithm of the function before graphing.
First, find the domain of the function. The scope includes all valid arguments of the function, that is, those arguments for which the function makes sense. It is clear that the denominator of the fraction cannot be zero, under the root cannot be a negative number. The base of the logarithm must be positive and not equal to one. The expression under the logarithm must be positive. Restrictions on the domain of the function can be imposed and the condition of the problem.
Consider how the domain of the function affects the set of values that a function can take.
The set of values of a linear function represents the set of all real numbers (x belongs R), as a direct set of a linear equation, infinite.
In the case of a quadratic function find the value of the vertex of the parabola (x0=-b/a, y0=y(x0). If the branches of the parabola are directed upwards (a>0), then the set of values of the function are all y>y0. If the branches of the parabola are directed downwards (a<0), the range of functions defined by inequalities y
The set of values of the cubic function, the set of real numbers (x belongs R). In General, the range of any function with an odd exponent (5, 7, ...) is a field of real numbers.
The set of values of exponential function (y=a^x, where a is a positive number) is all numbers greater than zero.
To find the set of values of linear-fractional or fractional-rational function, you need to find the equations of the horizontal asymptotes. Find the value of x at which the denominator vanishes. Imagine how the schedule would look like. Sketch the graph. Based on this, define the set of values of the function.
The set of values of trigonometric functions sine and cosine are strictly limited. Sine and cosine on the module cannot exceed unity. But the value of tangent and cotangent can be anything.
If the problem is to find the set of values of the function on the given interval of values of the argument, consider the function specifically at this segment.
While many of the function values is useful to determine the intervals of monotonicity of the function - ascending & descending. This allows us to understand the behavior of the function.
The plotting functions (or at least sketch graphics will assist you to define a set of function values. Consider the type of function with which you are dealing (logarithmic, rational, trigonometric, linear, quadratic, etc.).
Advice 3: How to find the smallest root
For the solution of the quadratic equation and find the smallest root discriminant is calculated. The discriminant is equal to zero only if the polynomial has multiple roots.
You will need
- mathematical Handbook;
Bring the polynomial to a square equation of the form ax2 + bx + c = 0 where a, b and c are arbitrary real numbers, with a in no case must not equal 0.
Substitute the values of the resulting quadratic equation into the formula for calculating the discriminant. This formula is as follows: D = b2 - 4ac. In that case, if D is greater than zero, the quadratic equation will have two roots. If D equals zero, both the calculated root, will be not only real, but equal. Third option: if D is less than zero, the roots will be a complex number. Calculate the value of the roots: x1 = (-b + sqrt (D)) / 2a and x2 = (-b - sqrt (D)) / 2a.
To calculate roots of a quadratic equation can also use the following formula: x1 = (-b + sqrt (b2 - 4ac)) / 2a and x2 = (-b - sqrt (b2 - 4ac)) / 2a.
Compare the two calculated root: root with the smallest value is the desired value you.
Not knowing the roots of a quadratic trinomial, you can easily find their sum and product. Use the vieta theorem, according to which the sum of the square roots of the trinomial represented in the form x2 + px + q = 0, is equal to the second ratio, that is, p, but with opposite sign. The product of the roots corresponds to the value of the free term q. In other words, x1 + x2 = – p, and x1x2 = q. For example, given the following quadratic equation: x2 – 5x + 6 = 0. To start, lay 6, two multiplier, and so that the sum of these multipliers was equal to 5. If you picked up the values correctly, then x1 = 2, x2 = 3. Test yourself: 3x2=6, 3+2=5 (as required, 5 with the opposite sign, i.e. "plus").
Be careful not to make a mistake, placing signs!
The number with the sign "minus" is always less than positive. If I compare two negative values, then less of them will be the fact that the module is more.