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???
at the point of inflection the second derivative changes its sign
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)
it is just the point of inflection. it is not necessary that at f'(x)=0 there is always a maxima of minima.
The Higher-Order Derivative Test
- 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
- 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).
- 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
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