The function f(x)=(x-2)^4 has an inflection point at x=2.It is true or false.
true.
To determine if the function f(x) = (x-2)^4 has an inflection point at x=2, we need to examine the second derivative of the function.
The first step is to find the first derivative of f(x). Let's denote the first derivative as f'(x):
f'(x) = 4(x-2)^3
Next, we find the second derivative by taking the derivative of f'(x):
f''(x) = 12(x-2)^2
Now, we can evaluate f''(x) at x=2 to determine if there is an inflection point at that x-value:
f''(2) = 12(2-2)^2 = 12(0)^2 = 12(0) = 0
Since the second derivative f''(x) evaluates to 0 at x=2, it means the function changes concavity at x=2. In this case, there is a change from concave down to concave up. An inflection point occurs when the concavity changes.
Therefore, the statement "The function f(x)=(x-2)^4 has an inflection point at x=2" is false, as there is no concavity change at x=2.