The research of function helps not only with creation of a function graph, but sometimes allows to take useful information on function, without resorting to its graphic representation. It is so optional to build the schedule to find the smallest value of function on any given piece.
Instruction
1. Let equation functions y = be set by f (x). Function is continuous and defined on a piece [a; b]. It is necessary to find the smallest value of function on this piece. Let's consider, for example, the f (x) function = 3x² + 4x³ + 1 on a piece [-2; 1]. Our f(x) is continuous and defined on all numerical straight line, so and on the set piece.
2. Find the first derivative of function on a variable x: f' (x). In our case we will receive: f' (x) = 3*2x + 4*3x² = 6x + 12x².
3. Define points in which f' (x) is equal to zero or cannot be defined. In our example f' (x) exists for all x, we will equate it to zero: 6x + 12x² = 0 or 6x (1 + 2x) = 0. It is obvious that the work addresses in zero if x = 0 or 1 + the 2nd = 0. Therefore, f' (x) = 0 at x = 0, x =-0.5.
4. Define those which belong to the set piece [an among the found points; b]. In our example both points belong to a piece [-2; 1].
5. It was necessary to calculate values of function in points of zeroing of a derivative and also on the ends of a piece. The smallest of them will be the smallest value of function on a piece. Let's calculate values of function at x =-2,-0.5, 0 and 1.f (-2) = 3*(-2)² + 4*(-2)³ + 1 = 12 - 32 + 1 = - 19f (-0.5) = 3*(-0.5)² + 4*(-0.5)³ + 1 = 3/4 - 1/2 + 1 = 1.25f (0) = 3*0² + 4*0³ + 1 = 1f (1) = 3*1² + 4*1³ + 1 = 3 + 4 + 1 = 8takim image, the smallest value of the f (x) function = 3x² + 4x³ + 1 on a piece [– 2; 1] f(x) =-19 is, it is reached on the left end of a piece.