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The tangent to the parabola y=x^2 has be...

The tangent to the parabola `y=x^2` has been drawn so that the abscissa `x_0` of the point of tangency belongs to the interval [1,2]. Find `x_0` for which the triangle bounded by the tangent, the axis of ordinates, and the straight line `y=x0 2` has the greatest area.

Text Solution

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x+y =60
y=60-x
`x^(3)y=(60-x)x^(3)`
`x^(3)y=(60-x)x^(3)`
Let `f(x)=(60-x)x^(3),x in (0,60)`
Let `f(X)=3x^(2)(60-x)-x^(3)=0`
or x=45
`f(45^(+))lt0` and `f(45^(-))gt0`
Hence local maxima is at x=45
So , x= 45 y =15
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