How do you find the intervals of increasing and decreasing using the first derivative given #y=x+4/x#?

Question

Charles Arocha · Accepted Answer

The intervals of increasing are #x in ]-oo,-2[uu]2,+oo[#
The intervals of decreasing are #x in ]-2,0[uu]0,2[# We need #(1/x)'=-1/x^2# The domain of #y# is #D_y=RR-{0}# We figure out the initial derivative. #y=x+4/x# #dy/dx=1-4/x^2# To find the critical points, we calculate the values of #x# when #dy/dx=0# when #1-4/x^2=0# #1=4/x^2# #x^2=4# Therefore, #x=-2# and #x=2# We are able to construct the chart. #color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-2##color(white)(aaaaaaaa)##0##color(white)(aaaaaaa)##2##color(white)(aaaaa)##+oo# #color(white)(aaaa)##x+2##color(white)(aaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##||##color(white)(aaa)##+##color(white)(aaaa)##+# #color(white)(aaaa)##x-2##color(white)(aaaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aaa)##-##color(white)(aaaa)##+# #color(white)(aaaa)##dy/dx##color(white)(aaaaaaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aaa)##-##color(white)(aaaa)##+# #color(white)(aaaa)##y##color(white)(aaaaaaaaaa)##↗##color(white)(aaaa)##↘##color(white)(aaa)##||##color(white)(aaa)##↘##color(white)(aaaa)##↗# The intervals of increasing are #x in ]-oo,-2[uu]2,+oo[# The intervals of decreasing are #x in ]-2,0[uu]0,2[#

Evelyn Agard · Answer

undefined To find the intervals of increasing and decreasing using the first derivative $ \frac{dy}{dx} $ given $ y = x + \frac{4}{x} $: 1. Find the first derivative $ \frac{dy}{dx} $ of the given function $ y = x + \frac{4}{x} $. 2. Set the first derivative equal to zero and solve for $ x $ to find critical points. 3. Determine the intervals where the first derivative is positive to identify where the function is increasing. 4. Determine the intervals where the first derivative is negative to identify where the function is decreasing. Let's proceed with the steps: 1. Find the first derivative: $$ \frac{dy}{dx} = 1 - \frac{4}{x^2} $$ 2. Set $ \frac{dy}{dx} $ equal to zero: $$ 1 - \frac{4}{x^2} = 0 $$ $$ 1 = \frac{4}{x^2} $$ $$ x^2 = 4 $$ $$ x = \pm 2 $$ So, the critical points are $ x = 2 $ and $ x = -2 $. 3. Test intervals to find where the first derivative is positive or negative: - For $ x < -2 $: Choose $ x = -3 $, $ \frac{dy}{dx} = 1 - \frac{4}{(-3)^2} = -\frac{5}{9} < 0 $ - For $ -2 < x < 2 $: Choose $ x = 0 $, $ \frac{dy}{dx} = 1 - \frac{4}{0^2} $ is undefined, so we choose a value slightly above and below 0: - For $ x = 1 $, $ \frac{dy}{dx} = 1 - \frac{4}{1^2} = -3 < 0 $ - For $ x = -1 $, $ \frac{dy}{dx} = 1 - \frac{4}{(-1)^2} = 1 > 0 $ - For $ x > 2 $: Choose $ x = 3 $, $ \frac{dy}{dx} = 1 - \frac{4}{3^2} = \frac{5}{9} > 0 $ Based on these tests: - The function $ y = x + \frac{4}{x} $ is decreasing for $ x < -2 $ and $ -2 < x < 2 $. - The function $ y = x + \frac{4}{x} $ is increasing for $ x > 2 $.

Scarlett Allison · Answer

undefined To find the intervals of increasing and decreasing for the function $y = x + \frac{4}{x}$, you can use the first derivative test. Follow these steps: 1. Find the first derivative of the function $y = x + \frac{4}{x}$. 2. Set the first derivative equal to zero and solve for $x$ to find critical points. 3. Determine the intervals where the first derivative is positive and negative. 4. Use the sign of the first derivative to identify intervals of increasing and decreasing. Let's proceed with the calculations: 1. Find the first derivative of the function $y = x + \frac{4}{x}$: $$y' = 1 - \frac{4}{x^2}$$ 2. Set the first derivative equal to zero and solve for $x$ to find critical points: $$1 - \frac{4}{x^2} = 0$$ $$1 = \frac{4}{x^2}$$ $$x^2 = 4$$ $$x = \pm 2$$ So, the critical points are $x = 2$ and $x = -2$. 3. Determine the intervals where the first derivative is positive and negative: For $x < -2$, $y' > 0$, so the function is increasing. For $-2 < x < 2$, $y' < 0$, so the function is decreasing. For $x > 2$, $y' > 0$, so the function is increasing. 4. Use the sign of the first derivative to identify intervals of increasing and decreasing: The function is increasing on the intervals $(- \infty, -2)$ and $(2, + \infty)$. The function is decreasing on the interval $(-2, 2)$. These are the intervals of increasing and decreasing for the function $y = x + \frac{4}{x}$.