Prove that the function $ h(x) = x^3 + x^2 + x + 1 $ does not have any maximum or minimum.

Q: Prove that the function $ h(x) = x^3 + x^2 + x + 1 $ does not have any maximum or minimum.

Explanation: We have, \[ h(x) = x^3 + x^2 + x + 1 \Rightarrow h'(x) = 3x^2 + 2x + 1 \] Now, the discriminant of $ h'(x) = 3x^2 + 2x + 1 $ is: \[ \Delta = (2)^2 - 4(3)(1) = 4 - 12 = -8 0 \) for all $ x \in \mathbb{R} $. Therefore, the function is strictly increasing and has no maximum or minimum. Hence, $ h(x) = x^3 + x^2 + x + 1 $ does not have any local maxima or minima.

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Explanation: We have,
$$
h(x) = x^3 + x^2 + x + 1 \Rightarrow h'(x) = 3x^2 + 2x + 1
$$
Now, the discriminant of $ h'(x) = 3x^2 + 2x + 1 $ is:
$$
\Delta = (2)^2 - 4(3)(1) = 4 - 12 = -8 < 0
$$
So, the derivative has no real roots.  
This implies $ h'(x) \neq 0 $ for all $ x \in \mathbb{R} $.

Also, since the leading coefficient is positive, $ h'(x) > 0 $ for all $ x \in \mathbb{R} $.  
Therefore, the function is strictly increasing and has no maximum or minimum.

Hence, $ h(x) = x^3 + x^2 + x + 1 $ does not have any local maxima or minima.

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