Divide the number 4 into two positive number such that the sum of the square of one and the cube of other is minimum

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Please find below the solution to the asked query:

Let two numbers be x and y.We havex+y=4y=4-xLetS=x3+y2=x3+4-x2S=x3+x2-8x+16dSdx=3x2+2x-8For maxima or minima dSdx=03x2+2x-8=03x2+6x-4x-8=03xx+2-4x+2=03x-4x+2=0Either 3x-4=0 or x+2=0x=43 or x=-2dSdx=3x2+2x-8d2Sdx2=6x+2Putting x=43, we get,d2Sdx2=6.43+2=14>0As d2Sdx2>0 at x=43S will ne minimum when x=43y=4-43=12-43y=83Hence numbers are 43 and 83.

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