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Inflection Point Calculator
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An inflection point is a point on the curve. The curvature at this point changes its sign from positive to negative and vice versa.

Inflection Point Calculator
calculates the inflection point of the given curve if the function by which it is represented is given.
If f and f' are differentiable at point x then x is inflection point. Hence f"(x) = 0 and f'''(x) $\neq$ 0.

A function is given below as a default input for this calculator. When you click on "Calculate Inflection Point", first derivative of the function is calculated and then second derivative is calculated keeping it equal to zero. Value of x is substituted in the given function to get the inflection point. Finally, plot a graph at the point of inflection.

Steps for Inflection Point Calculator

Step 1 :

Observe the given function and find the first and second derivative of the given function.

Step 2 :

Equate the value of second derivative to zero to get the value of x. Now find the point of inflections by substituting the value of x in given functions.

The points of inflections are (x, f(x)).

Step 3 :

Plot the graph at the point of inflection.

Problems on Inflection Point Calculator

1. Find the inflection points of the given function where f(x) = 6x3.

Step 1 :

given: f(x) = 6x3
f'(x) = 18x2
f"(x) = 36x

Step 2 :

For f"(x) = 0,

36x = 0
The value of x is x = 0

Substituting the value of x, we get f(0) = 6(0)3 = 0

Step 3 :

The points of inflections are (0,0).

2. Find the inflection points of the given function where f(x) = x4 - 3 x3.

Step 1 :

given: f(x) = x4 - 3x3

f'(x) = 4x3 - 9x2.
f"(x) = 12x2 - 18x

Step 2 :

For f"(x) = 0
12x2 - 18x = 0
2x (6x - 9) = 0
The values of x are x = 0 and x = $\frac{3}{2}$

Substuting x = 0 in the given function, we get f(0) = 0.
The point of inflection are (0,0).

If x = $\frac{3}{2}$ in the given function, we get

f($\frac{3}{2}$) = ($\frac{3}{2})^{4}$ - 3 ($\frac{3}{2})^{3}$
= - 5.0625.

The point of inflection are (1.5, - 5.0625).

Step 3 :

The point of inflection are (0,0) and (1.5, - 5.0625).

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