Let f(x) be a polynomial function of second degree.

Question

Let $f(x)$ be a polynomial function of second degree. If $f(1)=f(-1)$ and $a,b,c$ are in A.P., then $f'(a),f'(b),f'(c)$ are in

Accepted Answer

$$f\left( x \right) = a{x^2} + bx + c$$

$$f\left( 1 \right) = f\left( { - 1} \right)$$

$$ \Rightarrow a + b + c = a - b + c$$

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 $A.P.$

Solutions · 1