The distance travelled by a particle from a fixed point is given as s = (t3 - t2 - t + 5)cm. Find the minimum distance that the particle can cover from the ...

The distance travelled by a particle from a fixed point is given as s = (t3 - t2 - t + 5)cm. Find the minimum distance that the particle can cover from the fixed point.

Answer Details

The distance travelled by the particle is given by the function s = t^3 - t^2 - t + 5. To find the minimum distance that the particle can cover from the fixed point, we need to find the minimum value of this function. Taking the derivative of s with respect to t, we get s' = 3t^2 - 2t - 1. Setting s' equal to zero and solving for t, we get t = (2 ± √10)/3. To determine whether this critical point is a minimum or maximum, we need to check the second derivative of s. Taking the derivative of s' with respect to t, we get s'' = 6t - 2. When t = (2 + √10)/3, we have s'' > 0, which means that this critical point corresponds to a minimum. Therefore, the minimum distance that the particle can cover from the fixed point is achieved when t = (2 + √10)/3. Substituting this value of t into the function s, we get s_min = s((2 + √10)/3) = (16 - 8√10 + 27√10 - 25)/27 = (-9 + 19√10)/27 ≈ 0.463 cm. Therefore, the correct option is not listed as it is approximately 0.463 cm.