Nature of Points

By
Can you write a function whose derivative does NOT exist a some point.🗨 🙋
🕵 🗩 Yes, below is a function fiven by f(x)=|x| whose derivative does NOT exist at x=0 .

Given function is
\(f(x)= |x|\)
In the simplified form, this function can be written as
\( f(x)= \begin{cases} -x &\text { for } x < 0 \\ x &\text { for } x>0 \end{cases} \)
The derivative function is given by
\( f'(x)= \begin{cases} -1 &\text { for } x < 0 \\ 1 &\text { for } x>0 \end{cases} \)
So,
At \( x=0^{-}\), the left hand limit is
\( \displaystyle \lim_{x \to 0^{-}} f'(x)= \lim_{x \to 0^{-} } -1= -1\) LHL
At \( x=0^{+}\), the right hand limit is
\( \displaystyle \lim_{x \to 0^{+}} f'(x)= \lim_{x \to 0^{+} } 1= 1\) RHL
Since, LHL≠RHL, so, the derivative of the function f(x)=|x| does NOT exist at x=0.

Nature of points

There are 6 types of points, they are describes as below.
  1. Critical Point

    A point where the derivative of a function is either zero or undefined, is called critical points.
    In the figure below, A,B,C, and D are critical points in which

    1. f'(x)=0 at A
    2. f'(x)=0 at B
    3. f'(x) is NOT defined at C (f is not differentiable at C)
    4. f'(x)=0 at D
    Critical points are the points on the graph where the function's rate of change is altered—either a change from increasing to decreasing, in concavity, or in some unpredictable fashion.
    A critical point is
    1. stationary point if the function changes from increasing/decreasign to decreasing/increasing at that point
    2. turning point if the function changes from increasing/decreasign to decreasing/increasing at that point and and is a local minimum/maximum
    3. saddle point if the function changes both increasing/decreasign and concavity at that point
    NOTE: Maxima and mimina can occurs at critical points where first derivative is undefined.
  2. Stationary Point

    A point where the derivative of a function (the gradient/slope of a graph) is zero is called a stationary point.
    In the figure below, A,B,and D are stationary points in which

    1. f'(x)=0 at A
    2. f'(x)=0 at B
    3. f'(x)=0 at D
    Note: all stationary points are critical points, but not all critical points are stationary points.
  3. Turning Point

    A point where the derivative of a function is (the gradient/slope of a graph) is zero and function has a local maximum or local minimum (extrema point), is called a turning point.There are two types of turning points: local maximum and local minimum.

    1. If f is increasing on the left interval and decreasing on the right interval, then the stationary point is a local maximum
    2. If f is decreasing on the left interval and increasing on the right interval, then the stationary point is a local minimum

    In the figure below, C is only the turning point[Please note that A and B is not turning point] in which

    1. f'(x)=0 at A, but local extrema is NOT defined at A
    2. f'(x) is NOT defined at B
    3. f'(x)=0 at C, local minima is defined at C
    Note: all turning points are stationary points, but not all stationary points are turning points.
  4. Straight line/Constant function

    Some stationary points are neither turning points nor horizontal points of inflection. For example, every point on the graph of the equation \(y = 1\) (see Figure below), or on any horizontal line, is a stationary point that is neither a turning point nor point of inflection. It is a straight line.

  5. Point of Inflection/ Plateau point

    A point where the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa, is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection.

    Mathematically, a point on a function f (x) is said to be a point of inflexion if f''(x) = 0 and f'''(x)≠0.
    At this point, the concavity changes from upward to downward or vice-versa.
    Therefore the point of inflexion is the transition between concavity of the curves.

  6. Saddle point (Horizontal point of inflection)

    A saddle point or minimax point is a point on a function where the slopes (derivatives) is zero , but which is not a local extremum of the function. Thus, a saddle point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum.

    Figure below shows the graph of the function \( f(x) = x^3\) .
    The derivative of this function is \( f'(x) = 3x^2\), so the 1st derivative f'(x) =0 when x = 0 is
    \(f'(0) = 3 \times 0^2 = 0\).
    So the function \( f(x) = x^3\) has a stationary point at x = 0.
    However, this stationary point isn’t a turning point.
    Because
    The second derivative of this function \( f''(x) = 6x\), so the second derivative \( f''(x) = 0\) when x = 0 is
    \(f''(0) = 6 \times 0 = 0\).
    So the function \( f(x) = x^3\) has a inflexional point at x = 0.
    In such case, the point is called saddle point

Find the critical points

  1. \(f(x)=8x^3+81x^2-42x-8\)

    Solution 👉 Click Here

  2. \(f(x)=1+80x^3+5x^4-2x^5\)

    Solution 👉 Click Here

  3. \(f(x)=2x^3-7x^2-3x-2\)
  4. \(f(x)=x^6-2x^5+8x^4\)
  5. \(f(x)=4x^3-3x^2+9x+12\)
  6. \(f(x)=\frac{x+4}{2x^2+x+8}\)

    Solution 👉 Click Here

  7. \(f(x)=\frac{1-x}{x^2+2x-15}\)

    Solution 👉 Click Here

  8. \(f(x)=\sqrt[5]{x^2-6x}\)

    Solution 👉 Click Here

Find the inflexional point

  1. Determine the concavity of \(f(x) = x^ 3 − 6 x^ 2 −12 x + 2\) and identify any points of inflection.

    Solution 👉 Click Here

Find the concavity

  1. Determine the concavity of \(f(x) = \sin x + \cos x\) on [0,2π] and identify any points of inflection

    Solution 👉 Click Here

  2. Discuss the curve/function with respect to cancavity, point of inflexion, and local maxima, minima.
    1. \(y=x^4-4x^3\)
    2. \(y=x^{2/3}(6-x)^{1/3}\)
    3. \( f(x) = \frac{1}{6}x^4-2x^3+11x^2-18x\)
    4. \( f(x) = \frac{1}{12}x^4-2x^2\)
    5. \( f(x) = x^3-3x^2+1\)
    6. \( f(x) = \frac{2}{3}x^3-\frac{5}{2}x^2+2x\)

Question for Practice

  1. Find the stationary points of the function \( f(x) = 2x^3-3x^2-36x\).
  2. Find the approximate values, to two decimal places, of the stationary points of the function \( f(x) = x^3-x^2-2x\).
  3. Find the (a) intervals of increasing/decreasing (b) intervals of concave upward/downward (c) point of inflection
    1. \(f(x)=2x^3-3x^2-12x\)
    2. \( f(x) = \frac{4}{3}x^3 + 5x^2- 6x-2\)
    3. \(f(x)=2+3x-x^3\)
    4. \( f(x) = 3x^4 -2x^3- 9x^2 + 7\)
    5. \(f(x)=x^4-6x^2\)
    6. \(f(x)=200+8x^3+x^4\)
    7. \(f(x)=3x^5-5x^3+3\)
    8. \(f(x)=(x^2-1)^3\)
    9. \(f(x)=x \sqrt{x^2+1}\)
    10. \(f(x)=x-3x^{1/3}\)
  4. Find the (a) vertical and horizental asymptotes (b) intervals of increasing/decreasing (c) intervals of concave upward/downward (d)point of inflection of following functions
    1. \(f(x)=\frac{1+x^2}{1-x^2}\)
    2. \(f(x)=\frac{x}{(x-1)^2}\)
    3. \(f(x)=\sqrt{x^2+1}-x\)

Find the local Extreme

  1. Find any turning points of the function \(f(x)=x^3-10x^2+25x+4\) and specify whether each turning point is a global maximum or minimum, or a local maximum or minimum.

    Solution 👉 Click Here

Question for Exam

  1. Suppose you are given formula for a function f
    1. How do you determine where f is increasing/decreasing
    2. How do you determine where f is concave upward/downward
    3. How do you locate inflexional points
  2. Find the critical numbers of \(f(x)=x^4(x-1)^3\).What does the second derivative tells you about the behavior of f at these points. What does the first derivative test tell you?

No comments:

Post a Comment