f(x) = 1/2 x^4 - x^2 +5
find the global maximum and the global minimum values of f(x) (if they exist) on [0,2]

please and thank you!

Question

f(x) = 1/2 x^4 - x^2 +5
find the global maximum and the global minimum values of f(x) (if they exist) on [0,2]

please and thank you!

anonymous · Answer

Procedure:
Find stationary points, that is points at which the gradient is zero i.e. f'(x)=0 such that x is in [0,2]
Determine for each of these whether they are maxima or minima through use of the second derivative.

Differentiating:
$$f'(x) = 2x^3-2x$$

Solving for f'(x)=0 we get x=0,1 in [0,2]
$$f''(x) = 6x^2-2$$

f''(0) = -2 < 0 implies that 0 is a maximum point.
f''(1) = 4 > 0 implies that 1 is a minimum point.

We are done.  Ask for clarification is necessary.

anonymous · Answer

thank you so much!