Exams Joint Entrance Examination Mathematics Application of Derivatives Question 162 Question 162 of 212 Medium MCQ
If Rolle's theorem holds for the function f ( x ) = x 3 − a x 2 + b x − 4 f(x) = {x^3} - a{x^2} + bx - 4 f ( x ) = x 3 − a x 2 + b x − 4 , \[x \in [1,2]\] with \[f'\left( {{4 \over 3}} \right) = 0\], then ordered pair (a, b) is equal to :
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f ( 1 ) = f ( 2 ) f(1) = f(2) f ( 1 ) = f ( 2 )
\[ \Rightarrow 1 - a + b - 4 = 8 - 4a + 2b - 4\]
3 a − b = 7 3a - b = 7 3 a − b = 7 ..... (1)
f ′ ( x ) = 3 x 2 − 2 a x + b f'(x) = 3{x^2} - 2ax + b f ′ ( x ) = 3 x 2 − 2 a x + b
\[ \Rightarrow f'\left( {{4 \over 3}} \right) = 0 \Rightarrow 3 \times {{16} \over 9} - {8 \over 3}a + b = 0\]
\[ \Rightarrow - 8a + 3b = - 16\] ..... (2)
\[ \therefore \] a = 5 , b = 8 a = 5,b = 8 a = 5 , b = 8 Answer the question to unlock