# Application of derivative in investigation of functions*Continuity and differentiability of function. Sufficient * conditions of functions monotony. Darboux's theorem. Intervals of function monotony. Critical points. Extreme ( minimum, maximum ). Points of extreme. Necessary condition of extreme. Sufficient conditions of extreme. Plan of function investigation. *Relation between continuity and differentiability of function. **If a function * *is differentiable at some point, then it is a continuous function at this point. Contrary is invalid: a continuous function can have no derivative.* C o n s e q u e n c e .* If a function is discontinuous at some point,**then it has* *no derivative in this point.*
E x a m p l e . | The function *y* = | *x* | ( Fig.3 ) is continuous everywhere, but it has no derivative at *x* = 0 , because a tangent of the graph at this point does not exist. ( think, please, why ? ) | *Sufficient conditions of functions monotony. *
*If f *’(* x *)* > 0 at every point of an interval *(* a, b *)* , then a function f *(* x *)* **increases within this interval.*
*If f *’(* x *)* < 0 at every point of an interval *(* a, b *)* , then a function f *(* x *)* **decreases within this interval.*
*Darboux’s theorem. **Points, at which a derivative of a function is equal to 0 * *or doesn’t exist, divide a function domain for such intervals that within each * *of them a derivative saves a constant sign.*
Using these signs it is possible to find *intervals of monotony* of functions, what is very important in investigations of functions.
Hence, the function increases in the intervals ( - , 0 ) and ( 1, + ) and decreases in the interval ( 0, 1 ). The point *x* = 0 isn’t included in the function domain, but as *x* approaches 0 an item *x *^{ - 2 }increases unboundedly, therefore
* * the function also increases unboundedly. At the point *x* = 1 the function value is 3. According to this analysis we can draw the graph of the function, represented on Fig.4*b *. *Critical points. *A domain interior points, in which a derivative of a function is equal to zero or doesn’t exist, are called *critical points *of this function. These points are very important at analysis and drawing a function graph, because only they can be points, in which a function has an *extreme* ( *minimum* or * maximum*, Fig.5*a, b* ).
At points *x*_{1 }, *x*_{2} ( Fig.5*a* ) and *x*_{3} ( Fig.5*b* ) a derivative is equal to 0; at points *x*_{1 }, *x*_{2}* * ( Fig.5*b* ) a derivative doesn’t exist. But all they are points of extreme. *Necessary condition of extreme.* *If x*_{0}* is an extreme point of a function f *(* x * ) * * *and* *a derivative f’ exists at this point, then f’ *(* x*_{0}* *)* = *0*.*
This theorem is a *necessary* condition of extreme. If a derivative of a function at some point is equal to zero, then it’s not necessarily, that the function has an extreme at this point. For instance, a derivative of the function *f* ( *x* ) = *x*³ is equal to 0 at* x* = 0 , but this function has no extreme at this point ( Fig.6 ). On the other hand, the function *y* = | *x* | , represented on Fig.3, has a minimum at the point *x* = 0 , but there is no derivative at this point. *Sufficient conditions of extreme. **If a derivative changes its sign from plus to * *minus at a point x*_{0}* , then x*_{0}* is a point of maximum. If a derivative changes its * *sign from minus to plus at a point x*_{0}* , then x*_{0}* is a point of minimum. *
*Plan of function investigation. *To draw a graph of a function it is necessary:
1) to find a domain and a codomain of a function, 2) to ascertain if the function is even or odd, 3) to determine if the function is periodic or not, 4) to find zeros of the function and its values at *x* = 0, 5) to find intervals of a sign constancy, 6) to find intervals of monotony, 7) to find points of extreme and values of a function in these points, 8) to analyze the behavior of a function near “special” points and at a great modulus of *x *. E x a m p l e . Analyze the function *f *( *x* ) = *x*³ + 2*x*² - *x* - 2 and draw its graph. S o l u t i o n . We’ll investigate the function by the above represented scheme.
1) a domain *x* * R * ( i.e. *x* – any real number ); a codomain *y* * R*, because *f* ( *x* ) is an odd degree polynomial; 2) *f* ( *x* ) is neither an even nor an odd function ( explain, please ); 3) *f* ( *x* ) is not a periodic function ( prove this, please, yourself ); 4) a graph of this function is intersected with * y*-axis in the point ( 0, - 2 ), because *f* ( 0 ) = - 2 ; to find zeros of the function it is necessary to solve the equation: *x*³ + 2*x*² - *x* - 2 = 0 , one of roots of which ( *x* = 1 ) is obvious. Other roots can be found ( if they exist ! ) by solving the quadratic equation: *x*² + 3*x* + 2 = 0, which is received after dividing the polynomial
* x*³ + 2*x*² - *x* - 2 by the binomial ( *x* – 1 ). It is easy to check, that the two other roots are: *x*_{2} = -2 and *x*_{3} = -1. So, zeros of this function are: * * -2, -1 and 1. 5) It means, that a numerical line is divided by these roots into four intervals of a sign constancy, inside of which the function saves its sign :
This result can be received from the polynomial factorization: *x*³ + 2*x*² - *x * - 2 = ( *x* + 2 ) ( *x* + 1 ( *x* – 1 ) and estimating the product sign by the * method of intervals. * 6) The derivative: *f’* ( *x* ) = 3*x*² + 4*x *-1 has no points, at which it doesn’t exist, therefore its domain is ** ***R* ( all real numbers ); zeros of * f’* ( *x* ) are roots of the equation: 3*x*² + 4*x *- 1 = 0 . The received results are included in the following table: Back |