Let $$f\left( x \right)$$ be a polynomial function of second degree. If $$f\left( 1 \right) = f\left( { - 1} \right)$$ and $$a,b,c$$ are in $$A.P, $$ then $$f'\left( a \right),f'\left( b \right),f'\left( c \right)$$ are in
A
Arithmetic -Geometric Progression
B
$$A.P$$
C
$$G.P$$
D
$$H.P$$
Explanation
$$f\left( x \right) = a{x^2} + bx + c$$
$$f\left( 1 \right) = f\left( { - 1} \right)$$
$$ \Rightarrow a + b + c = a - b + c$$
or $$b = 0$$
$$\therefore$$ $$f\left( x \right) = a{x^2} + c$$
or $$f'\left( x \right) = 2ax$$
Now $$f'\left( a \right);f'\left( b \right);$$
and $$f'\left( c \right)$$ are $$2a\left( a \right);2a\left( b \right);2a\left( c \right)$$
i.e.$$\,2{a^2},\,2ab,\,2ac.$$
$$ \Rightarrow $$ If $$a,b,c$$ are in $$A.P.$$ then
$$f'\left( a \right);f'\left( b \right)$$ and
$$f'\left( c \right)$$ are also in $$A.P.$$
3
AIEEE 2003
MCQ (Single Correct Answer)
If $$f\left( x \right) = {x^n},$$ then the value of
If $$f\left( y \right) = {e^y},$$ $$g\left( y \right) = y;y > 0$$ and
$$F\left( t \right) = \int\limits_0^t {f\left( {t - y} \right)g\left( y \right)dy,} $$ then