A particle moves along the x-axis so that at time t = 0 its position is given by x(t)=t^(5)-5t^(4). Determine all intervals when the particle is moving to the right. " Moving right "rarr v(t) 0

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A particle moves along the  x-axis so that at time  t >= 0 its position is given by  x(t)=t^(5)-5t^(4). Determine all intervals when the particle is moving to the right.  " Moving right "rarr v(t) > 0

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Find Derivative of Position Function: To determine when the particle is moving to the right, we need to find when its velocity, v(t), is greater than zero. The velocity is the derivative of the position function with respect to time. Let's find the derivative of x(t) = t^5 - 5t^4. v(t) = dx/dt = d/dt (t^5) - d/dt (5t^4) v(t) = 5t^4 - 20t^3Determine Velocity Greater Than Zero: Now we need to find the values of t for which v(t) > 0. Set the velocity function greater than zero and solve for t: 5t^4 - 20t^3 > 0Solve for Positive Velocity: Factor out the common term t^3: t^3(5t - 20) > 0Find Critical Points: Set each factor to zero to find potential critical points: t^3 = 0 and 5t - 20 = 0 Solving for t gives us t = 0 and t = 4.Test Intervals for Positive Velocity: We now have the critical points t = 0 and t = 4. We need to test the intervals created by these points to see where v(t) is positive. The intervals are (-∞, 0), (0, 4), and (4, ∞). However, since we are only considering t >= 0, we will test the intervals (0, 4) and (4, ∞).Test Interval (0, 4): Choose a test point in the interval (0, 4), say t = 2, and plug it into v(t): v(2) = 5(2)^4 - 20(2)^3 v(2) = 5(16) - 20(8) v(2) = 80 - 160 v(2) = -80 Since v(2) is negative, the particle is not moving to the right in the interval (0, 4).Test Interval (4, ∞): Choose a test point in the interval (4, ∞), say t = 5, and plug it into v(t): v(5) = 5(5)^4 - 20(5)^3 v(5) = 5(625) - 20(125) v(5) = 3125 - 2500 v(5) = 625 Since v(5) is positive, the particle is moving to the right in the interval (4, ∞).

A particle moves along the $x$ -axis so that at time $t \geq 0$ its position is given by $x(t)=t^{5}-5 t^{4}$ . Determine all intervals when the particle is moving to the right. $\newline$ \text { Moving right } \rightarrow v(t)>0

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A particle moves along the x x x-axis so that at time t≥0 t \geq 0 t≥0 its position is given by x(t)=t5−5t4 x(t)=t^{5}-5 t^{4} x(t)=t5−5t4. Determine all intervals when the particle is moving to the right.\newline \text { Moving right } \rightarrow v(t)>0

A particle moves along the $x$ -axis so that at time $t \geq 0$ its position is given by $x(t)=t^{5}-5 t^{4}$ . Determine all intervals when the particle is moving to the right. $\newline$ \text { Moving right } \rightarrow v(t)>0