91Ó°ÊÓ

To find the intervals where the function is increasing or decreasing, first find the derivative of the function \(f(x)\). Using the power rule, we get: $$f'(x) = -60x^{4} + 300x^{3} - 240x^{2}$$

Now we need to find the critical points of the function by setting the derivative \(f'(x)\) equal to zero, or determining where \(f'(x)\) is undefined. In this case, \(f'(x)\) is a polynomial, so it is always defined. We only need to find the zeros of the derivative: $$-60x^{4} + 300x^{3} - 240x^{2} = 0$$ Factor out the common factors: $$60x^2(-x^{2} + 5x - 4) = 0$$ Now, we solve the quadratic equation: $$-x^2 + 5x - 4 = 0$$ Using the quadratic formula: \(x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\), where \(a = -1\), \(b = 5\), and \(c = -4\): $$x = \frac{-5 \pm \sqrt{5^2 - 4(-1)(-4)}}{2(-1)}$$ $$x = \frac{-5 \pm \sqrt{9}}{-2}$$ So, \(x = 1\) or \(x = 4\). Thus, our critical points are \(x = 0\), \(x = 1\), and \(x = 4\).

Now, we need to analyze the intervals of the first derivative to see if the function \(f(x)\) is increasing or decreasing in each interval by using the first derivative test. The intervals we will test are defined by the critical points found in the previous step: - \((-\infty, 0)\) - \((0, 1)\) - \((1, 4)\) - \((4, \infty)\) Let's test one point in each interval and evaluate the sign of \(f'(x)\) at that point: 1. Interval \((-\infty, 0)\): Choose \(x = -1\), then \(f'(-1) = 60(1-15+16) = 120 > 0\). So, \(f(x)\) is increasing on \((-\infty, 0)\). 2. Interval \((0, 1)\): Choose \(x = \frac{1}{2}\), then \(f'(\frac{1}{2}) = -60(\frac{1}{16}) + 300(\frac{1}{8}) - 240(\frac{1}{4}) = -60 + 150 - 240 = -150 < 0\). So, \(f(x)\) is decreasing on \((0, 1)\). 3. Interval \((1, 4)\): Choose \(x = 2\), then \(f'(2) = -60(16) + 300(8) - 240(4) = -960 + 2400 - 960 = 480 > 0\). So, \(f(x)\) is increasing on \((1, 4)\). 4. Interval \((4, \infty)\): Choose \(x = 5\), then \(f'(5) = -60(625)+300(125)-240(25) =-37500 +37500 -12000=-12000 < 0\). So, \(f(x)\) is decreasing on \((4, \infty)\). In conclusion, the function \(f(x)\) is increasing on the intervals \((-\infty, 0)\) and \((1, 4)\), and decreasing on the intervals \((0, 1)\) and \((4, \infty)\).