If f(x) = x^3 - 10x^2 + 200x - 10,then | vedclass

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(C) Given $f(x) = x^3 - 10x^2 + 200x - 10$.To determine the intervals of increase or decrease,we find the derivative $f'(x)$:$f'(x) = 3x^2 - 20x + 200

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$f(x)$ is increasing throughout the real line

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(C) Given $f(x) = x^3 - 10x^2 + 200x - 10$.To determine the intervals of increase or decrease,we find the derivative $f'(x)$:$f'(x) = 3x^2 - 20x + 200$.For the function to be increasing,we require $f'(x) > 0$. Consider the quadratic expression $3x^2 - 20x + 200$.The discriminant $D$ of this quadratic is $D = b^2 - 4ac = (-20)^2 - 4(3)(200) = 400 - 2400 = -2000$.Since $D  0)$,the quadratic $3x^2 - 20x + 200$ is always positive for all real values of $x$.Alternatively,by completing the square:$f'(x) = 3(x^2 - \frac{20}{3}x) + 200 = 3(x^2 - \frac{20}{3}x + \frac{100}{9} - \frac{100}{9}) + 200 = 3(x - \frac{10}{3})^2 - \frac{100}{3} + 200 = 3(x - \frac{10}{3})^2 + \frac{500}{3}$.Since $3(x - \frac{10}{3})^2 \ge 0$ for all $x \in \mathbb{R}$,it follows that $f'(x) \ge \frac{500}{3} > 0$ for all $x \in \mathbb{R}$.Therefore,$f(x)$ is strictly increasing throughout the real line.

If $f(x) = x^3 - 10x^2 + 200x - 10$,then

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