Consider the function f(x)=x^n for positive integer values of n.
(a) For what values of n does the function have a relative minimum at the origin?
(b) For what values of n does the function have a point of inflection at the origin?
for even values of n > 0 f has a min
for odd values of n > 1, inflection
To determine whether the function f(x) = x^n has a relative minimum or a point of inflection at the origin, we need to analyze the behavior of the function based on the value of n.
(a) Relative minimum at the origin:
A relative minimum occurs at the origin (x = 0) when the first derivative of the function is equal to zero, and the second derivative is positive or zero.
Let's start by finding the first and second derivatives of f(x).
First derivative: f'(x) = n*x^(n-1)
Second derivative: f''(x) = n*(n-1)*x^(n-2)
To determine the values of n for which f(x) has a relative minimum at the origin, we need to evaluate f'(x) and f''(x) at x = 0.
f'(0) = n*0^(n-1) = 0
f''(0) = n*(n-1)*0^(n-2) = 0
Since both f'(0) and f''(0) are equal to zero, we cannot determine the existence of a relative minimum at the origin solely based on these derivatives.
However, we can use an alternate approach. When n is even, the function f(x) = x^n is an even function. An even function is symmetric about the y-axis, meaning it has a relative minimum at the origin. Therefore, for even values of n, f(x) has a relative minimum at the origin.
In conclusion, the values of n for which f(x) = x^n has a relative minimum at the origin are even positive values of n.
(b) Point of inflection at the origin:
A point of inflection occurs at the origin when the second derivative of the function is equal to zero.
We have already calculated the second derivative as f''(x) = n*(n-1)*x^(n-2).
To determine the values of n for which f(x) has a point of inflection at the origin, we set f''(0) = 0.
n*(n-1)*0^(n-2) = 0
Since any value multiplied by 0 is equal to 0, we can't determine a specific value of n for which the second derivative equals zero. Therefore, the function f(x) = x^n does not have a point of inflection at the origin for any positive value of n.
In summary, the values of n for which f(x) = x^n has a relative minimum at the origin are even positive values of n. The function does not have a point of inflection at the origin for any positive value of n.