 Hello friends and how are you all doing today? My name is Priyanka and the question says for the curve y is equal to 4x cube minus 2x5 Find all the points at which tangent passes through the origin We are given the equation of the curve as 4 sorry y is equal to 4x cube minus 2x raised to the power 5 on differentiating y With respect to x we get dy by dx is equal to 12x square minus 10x raised to the power 4 right Here let the point the x1 y1 so dy by dx At the points x1 y1 is equal to 12 x1 square minus 10 x1 raised to the power 4 now equation the tangent has at x1 y1 y minus y1 equal to slope That is 12 x1 square minus 10 x1 4 into x minus x1 Right now this tangent Passes through origin right this is given to us that means the point is 0 0 through which it passes So we have the value of x and y both equal to 0. So we have 0 minus y1 equal to 12 x1 square minus 10 x1 raised to the power 4 0 minus x1 This implies minus y1 is equal to minus x1 to 12 x 1 raised to the power 2 minus 10 x1 raised to the power 4 now When x is equal to x1 then the value of y was 4 x1 q minus 2 x1 raised to the power 5 we have substituted x1 sorry x as x1 so the value of y on substituting x as x1 Becomes 4 x1 q minus 2 x1 5. This is what we have written over here so on substituting y1 Let this equation be 1 let this equation be 2 from 1 in 2 We get 4 x1 q minus 2 x1 raised to the power 5 equal to on opening the brackets. We have 12 x1 q plus 10 x1 raised to the power 5 on simplifying we have 16 x1 q minus 12 x1 raised to the power 5 equal to 0 this implies Taking 4 x1 q common. We are left with 4 x1 square minus 3 x1 sorry On taking 4 x1 q common. We are left with 4 minus 3 x1 square in the bracket So this means either 4 x1 q is equal to 0 or 4 minus 3 x1 square is equal to 0 this implies x1 is equal to 0 or x1 is equal to 2 upon root 3 now when x1 is equal to 0 this means y1 is equal to 0 and When x1 is equal to 2 by root 3 the value of y1 is equal to 4 on 2 root 3 The whole cube minus 2 of into 2 root 3 the whole raised to the power 5 Which is 4 into 8 upon 3 root 3 minus 2 into 32 upon 9 root 3 which is 32 upon 3 root 3 minus 64 upon 9 root 3 9 root 3 32 into 3 Will give us 96 minus 64 So the value is 32 upon 9 root 3 so the answer is The value is 0 comma 0 or it is 2 root 3 comma 32 upon 9 root 3 Right this completes the session. Hope you understood it. Have a nice day