The equation of a curve is given by $$y=f(x)$$ wheref′(x) is a continuous function. The tangent at $$points(1$$, $$f(1))$$ , (2$$,$$f(2)) and (3$$,$$f(3)) make angles π$$6$$,π$$3and$$ π$$4$$ respectively with positive $$x-axis$$. Then∫$$23$$ f′$$(x)f$$′′$$(x)dx+$$∫$$13$$ f′′$$(x)dx$$ is

Question

The equation of a curve is given by $$y=f(x)$$ wheref′(x) is a continuous function. The tangent at $$points(1$$, $$f(1))$$ , (2$$,$$f(2)) and (3$$,$$f(3)) make angles π$$6$$,π$$3and$$ π$$4$$ respectively with positive $$x-axis$$. Then∫$$23$$ f′$$(x)f$$′′$$(x)dx+$$∫$$13$$ f′′$$(x)dx$$ is

Infinity Learn · Accepted Answer

∫$$23$$ f′(x)f$$′′$$(x)dx+$$∫$$13$$ f′′$$(x)dx=f$$′$$(x)2223+f$$′$$(x)13=12f$$′$$(3)2$$−f′$$(2)2+f$$′$$(3)−f′$$(1)=12tan$$ π$$42$$−$$tan$$ π$$32+$$$$tan$$ π$$4$$−$$tan$$ π$$6=12$$[$$1$$−$$3$$]$$+1$$−$$13=$$−$$13$$

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The equation of a curve is given by y=f(x) wheref′(x) is a continuous function. The tangent at points(1, f(1)) , (2,f(2)) and (3,f(3)) make angles π6,π3and π4 respectively with positive x-axis. Then∫23 f′(x)f′′(x)dx+∫13 f′′(x)dx is

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