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Question

Let f be a differentiable function on R defined by f(x)=5(x+1)2. Let A be the point of intersection where the tangent line drawn to the graph of y=f(x) at the point P(x,f(x)) intersects with x-axis and B be the intersection point where the tangent line at P(x,f(x))intersects with y-axis. If S(x) denotes the area of ΔAOB, where O is the origin, then the minimum value of S(x) in the interval [0,1] is equal to pq, p and q being relatively prime. The value of p+q is

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Solution


Equation of tangent at P(x,f(x)) is
Yf(x)=f(x)(Xx)
So, A(xf(x)f(x),0)
and B(0,f(x)xf(x))

Let S(x) be the area of ΔAOB.
S(x)=12×OA×OB
S(x)=12[xf(x)f(x)][f(x)xf(x)]
=12[f(x)xf(x)]2f(x)
=[5x22x1+2x2+2x]24(x+1)
=(x2+4)24(x+1)

S(x)=2×4(x+1)(x2+4)2x4(x2+4)216(x+1)2
Putting S(x)=0, we get
3x2+4x4=0
(x+2)(3x2)=0
x=23 (0x1)
S(23)=8027, S(0)=4,S(1)=258
Minimum value of S(x) in [0,1] is 8027
Hence, p+q=80+27=107

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