Find the local maxima and local minima, if any, of the function $ f(x) = x^2 $. Find also the local maximum and the local minimum values.

Q: Find the local maxima and local minima, if any, of the function $ f(x) = x^2 $. Find also the local maximum and the local minimum values.

Explanation: We have: \[ f(x) = x^2 \Rightarrow f'(x) = 2x \] Setting $ f'(x) = 0 \Rightarrow x = 0 $ Also, \[ f''(x) = 2 > 0 \] So, $ x = 0 $ is a point of local minima. Local minimum value is: \[ f(0) = 0 \] There is no local maximum.

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Explanation: We have:
$$
f(x) = x^2 \Rightarrow f'(x) = 2x
$$
Setting $ f'(x) = 0 \Rightarrow x = 0 $

Also,
$$
f''(x) = 2 > 0
$$
So, $ x = 0 $ is a point of local minima.

Local minimum value is:
$$
f(0) = 0
$$

There is no local maximum.

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