$$ f(x)=\frac{x^{4}}{4}-\frac{x^{3}}{3}-3 x^{2}+10 $$ on $$ [-4,4] $$

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To find the critical points of the function $$ f(x) = \frac{x^{4}}{4} - \frac{x^{3}}{3} - 3x^{2} + 10 $$ on the interval $$[-4, 4]$$, we need to find the values of $$ x $$ where the derivative of the function is equal to zero or undefined.

First, let's find the derivative of the function $$ f(x) $$ with respect to $$ x $$ using the power rule for differentiation:

$$ f'(x) = \frac{d}{dx} \left( \frac{x^{4}}{4} - \frac{x^{3}}{3} - 3x^{2} + 10 \right) $$

$$ f'(x) = \frac{4x^{3}}{4} - \frac{3x^{2}}{3} - 6x $$

$$ f'(x) = x^{3} - x^{2} - 6x $$

Now, we need to find the critical points by setting the derivative equal to zero and solving for $$ x $$:

$$ x^{3} - x^{2} - 6x = 0 $$

$$ x(x^{2} - x - 6) = 0 $$

$$ x(x - 3)(x + 2) = 0 $$

Therefore, the critical points are $$ x = 0 $$, $$ x = 3 $$, and $$ x = -2 $$.

Next, we need to determine the nature of these critical points by examining the sign of the derivative in the intervals defined by these points. This will help us identify the local maxima and minima of the function on the interval $$[-4, 4]$$.
The critical points of the function $$ f(x) = \frac{x^{4}}{4} - \frac{x^{3}}{3} - 3x^{2} + 10 $$ on $$[-4, 4]$$ are at $$ x = 0 $$, $$ x = 3 $$, and $$ x = -2 $$.

\( f(x)=\frac{x^{4}}{4}-\frac{x^{3}}{3}-3 x^{2}+10 \) on \( [-4,4] \)

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