The function f(x)=3x^{4}+16x^{3}-30x^{2}+10 i | vedclass

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(C) Given function: $f(x)=3x^{4}+16x^{3}-30x^{2}+10$ Find the derivative: $f'(x)=12x^{3}+48x^{2}-60x$ For the function to be increasing,we require $f'

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$x \in(-5,0) \cup(1, \infty)$

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(C) Given function: $f(x)=3x^{4}+16x^{3}-30x^{2}+10$ Find the derivative: $f'(x)=12x^{3}+48x^{2}-60x$ For the function to be increasing,we require $f'(x) > 0$: $12x^{3}+48x^{2}-60x > 0$ Factor out $12x$: $12x(x^{2}+4x-5) > 0$ Factor the quadratic expression: $12x(x+5)(x-1) > 0$ To find the intervals,we identify the critical points: $x = -5, 0, 1$. Testing the intervals: For $x \in (-\infty, -5)$,$f'(x) For $x \in (-5, 0)$,$f'(x) > 0$. For $x \in (0, 1)$,$f'(x) For $x \in (1, \infty)$,$f'(x) > 0$. Thus,the function is increasing for $x \in (-5, 0) \cup (1, \infty)$.

The function $f(x)=3x^{4}+16x^{3}-30x^{2}+10$ is increasing for

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