Share

Books Shortlist

Solution for If F (X) = Ax2 + Bx + C is Such that F (A) = F (B), Then Write the Value of C in Rolle'S Theorem ? - CBSE (Science) Class 12 - Mathematics

ConceptMaximum and Minimum Values of a Function in a Closed Interval

Question

If f (x) = Ax2 + Bx + C is such that f (a) = f (b), then write the value of c in Rolle's theorem ?

Solution

We have

$f\left( x \right) = A x^2 + Bx + C$

Differentiating the given function with respect to x, we get

$f'\left( x \right) = 2Ax + B$

$\Rightarrow f'\left( c \right) = 2Ac + B$
$\therefore f'\left( c \right) = 0 \Rightarrow 2Ac + B = 0 \Rightarrow c = \frac{- B}{2A} . . . \left( 1 \right)$

$\because f\left( a \right) = f\left( b \right)$

$\therefore A a^2 + Ba + C = A b^2 + bB + C$

$\Rightarrow A a^2 + Ba = A b^2 + bB$

$\Rightarrow A\left( a^2 - b^2 \right) + B\left( a - b \right) = 0$

$\Rightarrow A\left( a - b \right)\left( a + b \right) + B\left( a - b \right) = 0$

$\Rightarrow \left( a - b \right)\left[ A\left( a + b \right) + B \right] = 0$

$\Rightarrow a = b, A = \frac{- B}{\left( a + b \right)}$

$\Rightarrow \left( a + b \right) = \frac{- B}{A} \left( \because a \neq b \right)$

From (1), we have

$c = \frac{a + b}{2}$
Is there an error in this question or solution?
Solution for question: If F (X) = Ax2 + Bx + C is Such that F (A) = F (B), Then Write the Value of C in Rolle'S Theorem ? concept: Maximum and Minimum Values of a Function in a Closed Interval. For the courses CBSE (Science), CBSE (Commerce), CBSE (Arts), PUC Karnataka Science
S