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

Question

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

Infinity Learn · Accepted Answer

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 $$integral)=x55$$−$$x42+x33+3x01=15$$−$$12+13+3=9130$$

The area bounded by the curve $y = f (x) = x^{4} - 2 x^{3} + x^{2} + 3$ , x-axis and ordinates corresponding to
minimum of the function f(x) is

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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

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The area bounded by the curve $y = f (x) = x^{4} - 2 x^{3} + x^{2} + 3$ , x-axis and ordinates corresponding to
minimum of the function f(x) is