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.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 f'(x)=0 at A
 f'(x)=0 at B
 f'(x) is NOT defined at C (f is not differentiable at C)
 f'(x)=0 at D
 stationary point if the function changes from increasing/decreasign to decreasing/increasing at that point
 turning point if the function changes from increasing/decreasign to decreasing/increasing at that point and and is a local minimum/maximum
 saddle point if the function changes both increasing/decreasign and concavity at that point
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 f'(x)=0 at A
 f'(x)=0 at B
 f'(x)=0 at D
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.
 If f is increasing on the left interval and decreasing on the right interval, then the stationary point is a local maximum
 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
 f'(x)=0 at A, but local extrema is NOT defined at A
 f'(x) is NOT defined at B
 f'(x)=0 at C, local minima is defined at C

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.

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 viceversa.
Therefore the point of inflexion is the transition between concavity of the curves.

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
 \(f(x)=8x^3+81x^242x8\)
 \(f(x)=1+80x^3+5x^42x^5\)
 \(f(x)=2x^37x^23x2\)
 \(f(x)=x^62x^5+8x^4\)
 \(f(x)=4x^33x^2+9x+12\)
 \(f(x)=\frac{x+4}{2x^2+x+8}\)
 \(f(x)=\frac{1x}{x^2+2x15}\)
 \(f(x)=\sqrt[5]{x^26x}\)
Find the inflexional point
 Determine the concavity of \(f(x) = x^ 3 − 6 x^ 2 −12 x + 2\) and identify any points of inflection.
Find the concavity
 Determine the concavity of \(f(x) = \sin x + \cos x\) on [0,2π] and identify any points of inflection
 Discuss the curve/function with respect to cancavity, point of inflexion, and local maxima, minima.
 \(y=x^44x^3\)
 \(y=x^{2/3}(6x)^{1/3}\)
 \( f(x) = \frac{1}{6}x^42x^3+11x^218x\)
 \( f(x) = \frac{1}{12}x^42x^2\)
 \( f(x) = x^33x^2+1\)
 \( f(x) = \frac{2}{3}x^3\frac{5}{2}x^2+2x\)
Question for Practice
 Find the stationary points of the function \( f(x) = 2x^33x^236x\).
 Find the approximate values, to two decimal places, of the stationary points of the function \( f(x) = x^3x^22x\).
 Find the (a) intervals of increasing/decreasing (b) intervals of concave upward/downward (c) point of inflection
 \(f(x)=2x^33x^212x\)
 \( f(x) = \frac{4}{3}x^3 + 5x^2 6x2\)
 \(f(x)=2+3xx^3\)
 \( f(x) = 3x^4 2x^3 9x^2 + 7\)
 \(f(x)=x^46x^2\)
 \(f(x)=200+8x^3+x^4\)
 \(f(x)=3x^55x^3+3\)
 \(f(x)=(x^21)^3\)
 \(f(x)=x \sqrt{x^2+1}\)
 \(f(x)=x3x^{1/3}\)
 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
 \(f(x)=\frac{1+x^2}{1x^2}\)
 \(f(x)=\frac{x}{(x1)^2}\)
 \(f(x)=\sqrt{x^2+1}x\)
Find the local Extreme
 Find any turning points of the function \(f(x)=x^310x^2+25x+4\) and specify whether each turning point is a global maximum or minimum, or a local maximum or minimum.
Question for Exam
 Suppose you are given formula for a function f
 How do you determine where f is increasing/decreasing
 How do you determine where f is concave upward/downward
 How do you locate inflexional points
 Find the critical numbers of \(f(x)=x^4(x1)^3\).What does the second derivative tells you about the behavior of f at these points. What does the first derivative test tell you?
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