(b) Verify Rolle's theorem for Cany...
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(b) Verify Rolle's theorem for Cany f(x) = 2x3 + x² - 4x-2

11th - 12th Class
Maths
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147
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according to Rolle's theorem. for a function ( f:[a, b] rightarrow mathbb{R} ) if (a) ( mathrm{f} ) is continuous in ( [mathrm{a}, mathrm{b}] ) (b) ( f ) is differentiable in ( (a, b) ) ( f(a)=f(b) ) Then there exists some ( c ) in ( (a, b) ) such that ( f^{prime}(c)=0 ) Let's check all conditions: as ( f(x)=x^{2}+2 x-8 ) is a polynomial function. ( f(x) ) is continuous for all real value of ( x ) hence, ( f(x) ) is continuous in [-4,2] we know, every polynomial function are differentiable. e.g., ( f(x)= ) ( 2 x+2 ) hence, ( f(x) ) is differentiable in [-4,2] now, ( f(-4)=(-4)^{2}+2(-4)-8 ) ( =16-8-8=16-16=0 ) ( f(2)=(2)^{2}+2(2)-8 ) ( =4+4-8=8-8=0 ) hence, ( f(-4)=f(2) ) hence, a point c exists in (-4,2) in such that ( f^{prime}(c)=0 ) because ( f^{prime}(x)=2 x+2 ) put ( x=c, f^{prime}(c)=2 c+2=0 ) ( c=-1 ) Rolle's theorem is verified
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