f(x)=x^n , where n is a positive integer greater or equal to 2. The graph of f(x) will have an inflection point when n is
A. Even
B. Odd
C. Divisible by 3
D. For all values
E. For no values
compare the graphs of x^2, x^3, x^4, x^5
Or, just consider the fact that an inflection point occurs when f"=0 and f' does not change sign.
To determine when the graph of the function f(x) = x^n has an inflection point, we need to analyze the behavior of the function's second derivative.
First, let's find the first and second derivatives of f(x) with respect to x.
f(x) = x^n
To find the first derivative, we apply the power rule for differentiation:
f'(x) = n*x^(n-1)
Now, let's find the second derivative by differentiating f'(x):
f''(x) = d/dx [n*x^(n-1)]
Using the power rule again, we differentiate and obtain:
f''(x) = n*(n-1)*x^(n-2)
An inflection point occurs when the concavity of the function changes, meaning when the second derivative changes signs.
If we look at the expression for f''(x), we can notice that its sign depends on the coefficient n(n-1) and the value of x. The sign of f''(x) will not change for all values of x unless the coefficient n(n-1) is zero.
So, let's consider the possibilities for n:
A. Even: If n is even, then n(n-1) will always be divisible by 2, meaning it will not be zero. Therefore, f(x) = x^n will not have any inflection points.
B. Odd: If n is odd, then n(n-1) will always be the product of two consecutive integers, meaning it will always be an even number. Again, f(x) = x^n will not have any inflection points.
C. Divisible by 3: If n is divisible by 3, it means that n(n-1) is divisible by 3 as well. However, since it is still divisible by 2, it is not zero. Therefore, f(x) = x^n will not have any inflection points.
D. For all values: This statement is incorrect since we have already established that f(x) = x^n will not have any inflection points.
E. For no values: This statement is correct. The graph of f(x) = x^n will not have any inflection points for any values of n.
Therefore, the answer is E. For no values.