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The maximum of f'(x) can be found when the gradient is 0. It is said to be a turning point, and its shape is concave up. When f'(x) intersects the x-axis, roots of the curve are formed. In increasing functions, there are positive gradients, and decreasing, the opposite. There is a non-stationary point of inflection, causing its tangent to be inflecting. The global maximum is the maximum value of y on the entire domain, and the global minimum is the minimum value of y on the entire domain.

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When the red function's f''(x) = 0, the green function y value is 0 and the blue function' f'(x) = 0. Furthermore, when the red function's gradient is > 0, all blue function y values are greater than 0. When all red function's gradient is < 0, the y values for the blue function are less than 0.

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Mathematics, a branch of human understanding that has improved society and wellbeing. This image elegantly displays the first and second derivatives of the function f(x). The simplistic, intuitive graph allows its viewers to acquire an understanding how the rates of functions change with respect to a variable x. The f '(x) function displays whether the function is increasing or decreasing as whenever the function is increasing, f '(x) will be positive. Hence, by gaining an understanding of the first derivative we can also locate local maximum and minimum points in the function by identifying the zeroes of the first derivative. With the following information, sign diagrams can also be drawn to find out the intervals where the function is increasing or decreasing. A negative to positive change represents a local maximum and a positive to negative change represents a local minimum.  The image is very effective in showing this to the viewers and thus is a good learning resource provided by Ms. Rahmat. Lastly, we can identify the inflection points in the function by looking at the maximum and minimum points of the first derivative graph, a handy observation when required to draw a graph from its derivatives. Now moving on to the second derivative, it can be seen that it provides some very useful information that we can use to plot various aspects of different graphs. Here, the second derivative function is linear and is easy to understand in respect to the other functions. Taking the second derivative of the function gives us where the original function is concave up and concave down. When the sign diagram of the second derivative is drawn, If the second derivative is positive, the graph is concave up, whereas if the derivative is negative, the graph is concave down for that interval. The x-value where the sign diagram of the second derivative is 0 is a point of inflection. This allows us to understand the rate of the first derivative.I would like to conclude by saying that Calculus is an integral part of understanding mathematics and therefore can be used as a vessel to improve our current society.
- Rishab

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The graph is concave down when f''(x) is < 0, and is concave up when f''(x) is > 0.
When f''(x) = 0, the point of non-stationary inflection of f(x) is present.

When the gradient of f(x) =0, f(x) is at a local maximum/minimum point.

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When f'(x) =0, f(x) is at a local maximum/minimum point.
Negative values of the sign diagram = original function decreasing. Positive values of sign diagram = original function increases.

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The graphs of derivatives are a fascinating mathematic concept. The first derivative is the collection of all tangents at points on a graph (f(x)). The second derivative reflects on the concavity of the graph f(x). If the graph f(x) is concave down for the domain differed by the point of inflection, the second derivative is smaller than 1, and negative. Conversely, if the selected portion of the graph is concave up—it is a parabola with gradients positive, the second derivative function is larger than 1 and positive. Additionally, the point of inflection on f(x) is the minimum or maximum value of f'(x) and the zero of f"(x). The stationary points of f(x), where the graph is at the local minima or maxima or a stationary inflection point, the x value of these coordinates are the zeros of f'(x). These graphs must be examined carefully as these graphs do not determine the entire function at a point, but the first derivative represents the gradient of the tangent and the second derivative represents the concavity of the function. to find the exact function, the function has to be found with the gradient f'(x) and by inserting a representative coordinate on f(x). The derivative functions can be found through the method of chain rule, product rule and quotient rule. The derivatives are an essentially tool to expand the understanding the graph past the graph on surface, but the collection of the gradient of tangent and the concavity.

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The artwork before my eyes is simply a glimpse of the beauty that math has to offer. I am blown away by the intricacies evident in the complicatedly simple applications and interpretations as well as analyses and approaches that can be taken from this. When the derivate is taken from a function and equated to 0, the minimums and maximums of the original function can be found. The sign diagram of the first derivative also helps us find where the original function is increasing and decreasing. For positive values of the sign diagram, the original function is increasing whereas for negative values of the sign diagram, the original function is decreasing.  Taking the second derivative of the function gives us where the original function is concave up and concave down. When the sign diagram of the second derivative is drawn, the positive values are where the original function is concave up and the negative values are where the original function is concave down. The x-value where the sign diagram of the second derivative is 0, there is a point of inflexion.  SAAHIL

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The artwork before my eyes is simply a glimpse of the beauty that math has to offer. I am blown away by the intricacies evident in the complicatedly simple applications and interpretations as well as analyses and approaches that can be taken from this. When the derivate is taken from a function and equated to 0, the minimums and maximums of the original function can be found. The sign diagram of the first derivative also helps us find where the original function is increasing and decreasing. For positive values of the sign diagram, the original function is increasing whereas for negative values of the sign diagram, the original function is decreasing.  Taking the second derivative of the function gives us where the original function is concave up and concave down. When the sign diagram of the second derivative is drawn, the positive values are where the original function is concave up and the negative values are where the original function is concave down. The x-value where the sign diagram of the second derivative is 0, there is a point of inflexion.  SAAHIL

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These three beautiful graphs before me tell a great story. The first graph is of the original function, and it is a cubic function. The second graph is the first derivative function of the original function and gives us information about where the graph is increasing or decreasing. It is a quadratic parabola. The second derivative can be equated to 0 to find the points where the graph is flat i.e where the gradient is 0. These points can be added to a sign diagram and hence through substitution can be used to determine the intervals where the graph is increasing or decreasing. The second derivative function is obtained by differentiating the first derivative function. This gives us a linear function. This function can be equated to 0 to find the points of inflection of the graph and therefore the intervals where the graph is concave up or concave down. If the second derivative is positive, the graph is concave up, whereas if the derivative is negative, the graph is concave down for that interval.

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This is a graph displaying three different types of functions, f(x), f'(x), and f"(x). To sketch the graph of f'(x) the inflection point(s) would need to be calculated. In order to calculate the inflection point(s) the derivative of the f(x) would need to be found and then be equated to 0 which would then give us the x coordinate(s) of the inflection point(s). The x coordinates need to be substituted back into f(x) to find out the y value(s), this then gives us the inflection point(s).

With the following information, sign diagrams can be drawn to find out the intervals where the function is increasing or decreasing.

The second derivative is found out and equated to 0, to find out one of two things: 1. To find out if the function is stationery or non-stationery inflection and 2. To find out at what intervals the function is concave up and down.

If the second derivate of the function is positive at a particular interval then it is concave up, and if it is negative it is concave down.

- Vedeesha

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The graph is concave down when f''(x) is < 0, and is concave up when f''(x) is > 0.
When f''(x) = 0, the point of inflection of f(x) lies.

When f'(x) =0, f(x) is at a local maximum/minimum point. To determine whether it's a maximum or a minimum, look at the graph of f''(x). If f''(x) is negative, the point is a local maximum. If f''(x) is positive, the point is a local minimum.

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Added thing: When the function is increasing then the gradient, or the first derivative, is positive. Similarly, when the function is decreasing, the first derivative is negative. You're welcome.

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the x values of the max and min points of f(x) become the x-intercepts of the f'(x) graph.

also, the x value of the point of inflection of f(x) becomes the x-value of the x-intercept of second derivative graph.

when f(x) is decreasing, f'(x) is negative

when f(x) is positive, f'(x) is positive

when f''(x) is positive -> concave up

when f''(x) is negative -> concave down

- nirvaan :)

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First and second derivative - Graph