Inflection Points In Calculus: Analysis Of Function Behavior

f'(x)=0 & f”(x)<0

f(x) has a maximum

When the first derivative of a function f(x) is equal to zero, it means that there is a stationary point at x. It could either be a maximum or minimum point of the function f(x). However, if f”(x) is negative at the same point x, then it indicates that the function has a maximum value at x. Let’s understand this concept with the help of an example.

Suppose we have a function f(x) = x^3 – 6x^2 + 9x + 2. We calculate its first and second derivatives as follows:

f'(x) = 3x^2 – 12x + 9
f”(x) = 6x – 12

To find the stationary point(s) of the function f(x), we set f'(x) = 0 and solve for x:

3x^2 – 12x + 9 = 0
x^2 – 4x + 3 = 0
(x – 3)(x – 1) = 0

So, the stationary points are x = 1 and x = 3.

Next, we need to determine the nature of these stationary points using the second derivative of f(x). If f”(x) < 0 at a particular stationary point, then it is a local maximum. Otherwise, it is a local minimum. Let's evaluate f''(x) at x = 1 and x = 3. f''(1) = 6(1) - 12 = -6 (negative value) f''(3) = 6(3) - 12 = 12 (positive value) Therefore, x = 1 corresponds to a local maximum of the function f(x), while x = 3 corresponds to a local minimum of the function.

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