you are right... the speed of the beetle is not constant with time.

to find the time at which the speed is max and min you need to find the speed as a function of time first.

to do this let $\displaystyle s(t)$ denote speed of the beetle at time $\displaystyle t$.

then [s(t)]^2= (dx/dt)^2 + (dy/dt)^2

this will give you speed as a function of time. differentiate the expression of speed w.r.t time and set it equal to zero. you will get the times at which the speed reaches maximum and minimum.

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