The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to f is increasing or decreasing. In other words, the second derivative tells us the rate of change of the rate of change of the original function.

What does the second derivative tell you?

The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to f is increasing or decreasing. In other words, the second derivative tells us the rate of change of the rate of change of the original function.

What is the turning point of a quadratic equation?

The turning point of a graph (marked with a blue cross on the right) is the point at which the graph “turns around”. On a positive quadratic graph (one with a positive coefficient of x 2 x^2 x2), the turning point is also the minimum point.

Does a cubic function always have a turning point?

In particular, a cubic graph goes to −∞ in one direction and +∞ in the other. So it must cross the x-axis at least once. Furthermore, all the examples of cubic graphs have precisely zero or two turning points, an even number.

How do you find the maximum and minimum of differentiation?

How to Find Maximum and Minimum Points Using Differentiation ?

1. Differentiate the given function.
2. let f'(x) = 0 and find critical numbers.
3. Then find the second derivative f”(x).
4. Apply those critical numbers in the second derivative.
5. The function f (x) is maximum when f”(x) < 0.
6. The function f (x) is minimum when f”(x) > 0.

How do you find the turning point of a cubic function?

General method for sketching cubic graphs: Determine the x-intercepts by factorising ax3+bx2+cx+d=0 and solving for x. Find the x-coordinates of the turning points of the function by letting f′(x)=0 and solving for x. Determine the y-coordinates of the turning points by substituting the x-values into f(x).

What are maximum and minimum turning points?

A maximum turning point is a turning point where the curve is concave upwards, f′′(x)<0 f ′ ′ ( x ) < 0 and f′(x)=0 f ′ ( x ) = 0 at the point. A minimum turning point is a turning point where the curve is concave downwards, f′′(x)>0 f ′ ′ ( x ) > 0 and f′(x)=0 f ′ ( x ) = 0 at the point.

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