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Differential Calculus

Blazing goIITian

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13 Sep 2007 15:36:50 IST

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Maxima and Minima --- Plz help

None

Discuss about the turning point of the function

f(x) = x⁵_{I can t solve this one plz help. If an odd degree of derivative of a fucntion is zero then is it impossible to say about the turning point of a fuction???}

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Anshuman Johri

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13 Sep 2007 15:43:57 IST

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it is possible

at the point of inflection the second derivative changes its sign

f''(x)=20x^3

now at x=0 f''(x)=0
x<0 => f''(x)<0
x>0 => f''(x)>0
i.e. at x=0 the second derivative changes its sign

so x=0 is the turning point of f(x)

Swashata Ghosh

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13 Sep 2007 20:34:44 IST

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Experts plz help !!!!!!!!!!!!!!!!!!!

Swashata Ghosh

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14 Sep 2007 22:18:11 IST

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I couldn't understand anshuman. We will have two graph for x^5. So how can we say whether the turning point is Local maxima or minima???????

Anshuman Johri

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14 Sep 2007 22:28:02 IST

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the turning point is neither a local maxima nor a local minima.
it is just the point of inflection. it is not necessary that at f'(x)=0 there is always a maxima of minima.

Avinash Sharma

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15 Sep 2007 09:03:56 IST

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Hello swashata4iit ,

This type of points are called point of inflexion.

The higher-order derivative test is used to find maxima, minima, and points of inflection in an n^th degree polynomial's curve.

The Higher-Order Derivative Test

Let

f

be a differentiable function on the interval

I

and let

c

be a point on it such that

1. f ¢ (c) = f ¢¢ (c) = f ¢¢¢ (c) = ???.. = f ^(n-1) (c) = 0

2. f ⁿ (c) exists and is non-zero

Then,

if n is even

f ⁿ (x) < 0 Þ x = c is a point of local maximum
f ⁿ (x) > 0 Þ x = c is a point of local minimum

if n is odd Þ x = c is a point of inflection

An inflection point, or point of inflection (or inflexion) can be defined in any of the following ways:

a point on a curve at which the tangent crosses the curve itself.

a point on a curve at which the curvature changes sign. The curve changes from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa. If one imagines driving a vehicle along the curve, it is a point at which the steering-wheel is momentarily 'straight', being turned from left to right or vice versa.

a point on a curve at which the second derivative changes sign. This is very similar to the previous definition, since the sign of the curvature is always the same as the sign of the second derivative, but note that the curvature is not the same as the second derivative.

a point (x,y) on a function, f(x), at which the first derivative, f ¢(x), is at an extremum, i.e. a minimum or maximum. (This is not the same as saying that y is at an extremum).

See Figure 2 where function y = x³ is shown, rotated, with tangent line at inflection point of (0,0).

Note that since the first derivative is at an extremum, it follows that the second derivative, f ¢¢ (x), is equal to zero, but the latter condition does not provide a sufficient definition of a point of inflection. One also needs the lowest-order non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection. (An example of such a function is

y = x 4 - x

It follows from the definition that the sign of f '(x) on either side of the point (x,y) must be the same. If this is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.

Points of inflection can also be categorised according to whether f ' (x) is zero or not zero.

if f ' (x) is zero, the point is a stationary point of inflection, also known as a saddle-point

if f ' (x) is not zero, the point is a non-stationary point of inflection

In the Figure 3 function y = x⁴ - x has been plotted with tangent line at non-inflection point of (0,0).

An example of a saddle point is the point (0,0) on the graph y = x³. The tangent is the x-axis, which cuts the graph at this point (See Figure 1).

A non-stationary point of inflection can be visualised if the graph y = x³ is rotated slightly about the origin. The tangent at the origin still cuts the graph in two, but its gradient is non-zero.

Note that an inflection point is also called an ogee, although this term is sometimes applied to the entire curve

Swashata Ghosh

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15 Sep 2007 21:23:54 IST

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Thank u sir. This relly helped me a lot

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