The function f(x) = 2 + 4x2 + 6x4 + 8x6 hasonly one maximaonly one minimano maxima and minimamany maxima and minima

The function f(x) = 2 + 4x^2 + 6x^4 + 8x^6 is a polynomial function of even degree (6). The leading coefficient (8) is positive.

For even-degree polynomial functions, if the leading coefficient is positive, the function will open upwards, meaning the ends of the graph will go off to positive infinity. This means the function has a global minimum, but no global maximum.

However, to determine if the function has any local maxima or minima, we need to take the derivative of the function and set it equal to zero to find critical points.

f'(x) = 8x + 24x^3 + 48x^5

Setting f'(x) = 0 gives the solutions x = 0. This is the only critical point, and by the second derivative test, we can determine if it's a local maximum, minimum, or neither.

f''(x) = 8 + 72x^2 + 240x^4

f''(0) = 8, which is greater than 0, so x = 0 is a local minimum.

Therefore, the function f(x) = 2 + 4x^2 + 6x^4 + 8x^6 has one global minimum and one local minimum, but no maxima.