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QJEEAdv06Ma2022: JEE Advanced 2006 Mathematics - Class 12 JEEAdv06Ma2022 Solution: Let the definite integral be defined by the formula For more…

Let the definite integral be defined by the formula
∫abf(x)dx=b−a2(f(a)+f(b)).\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \right)} \right).} For more accurate result for
c∈(a,b),c \in \left( {a,b} \right), we can use ∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx=F(c)\int\limits_a^b {f\left( x \right)dx = \int\limits_a^c {f\left( x \right)dx + \int\limits_c^b {f\left( x \right)dx = F\left( c \right)} } } so
that for c=a+b2,c = {{a + b} \over 2}, we get ∫abf(x)dx=b−a4(f(a)+f(b)+2f(c)).\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 4}\left( {f\left( a \right) + f\left( b \right) + 2f\left( c \right)} \right).}

If f''\left( x \right) < 0\,\forall x \in \left( {a,b} \right) and cc is a point such that a < c < b, and
(c,f(c))\left( {c,f\left( c \right)} \right) is the point lying on the curve for which F(c)F(c) is
maximum, then f′(c)f'(c) is equal to

Multiple Choice Options

  • A f(b)−f(a)b−a{{f\left( b \right) - f\left( a \right)} \over {b - a}}
  • B 2(f(b))−f(a)b−a{{2\left( {f\left( b \right)} \right) - f\left( a \right)} \over {b - a}}
  • C 2f(b)−f(a)2b−a{{2f\left( b \right) - f\left( a \right)} \over {2b - a}}
  • D 00

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