The area bounded by the curve y=f(x)=x4−2x3+x2+3, x-axis and ordinates corresponding tominimum of the function f(x) is

f′(x)=4x3−6x2+2x=2x2x2−3x+1=2x(2x−1)(x−1).Since f is a differentiable function, so extremum points of f(x) we must have f′(x)=0 so x=0,1/2,1. Now f′′(x)=12x2−12x+2,f′′(0)=2,f′′(1)=2 and f′′(1/2)=3−6+2=−1Thus the function has minimum at x=0 and x=1. Therefore, the required area =∫01 x4−2x3+x2+3dx (See Chapter 14 for area as definite integral)=x55−x42+x33+3x01=15−12+13+3=9130