"Let for $f(x) = {a_0}{x^2} + {a_1}x + {a_2},\,f'(0) = 1$ and $f'(1) = 0$. If a0, a1, a2 are in an arithmatico-geometric progression, whose corresponding A.P. has common difference 1 and corresponding G.P. has common ratio 2, then f(4) is equal to _____________."

Question

Accepted Answer

"

Given,

$f(x) = {a_0}{x^2} + {a_1}x + {a_2}$

$f'(0) = 1$

$f'(1) = 0$

a₀, a₁, a₂ are in A. G. P

Common difference of $AP = 1$

Common ratio of $GP = 2$

A.P terms = a, a + 1, a + 2

G.P terms = y, ry, r²y

$\therefore$ AGP terms = ay, (a+1)ry, (a+2)r²y

$\therefore$ ${a_0} = ay$

${a_1} = (a + 1)ry = (a + 1)2y$

${a_2} = (a + 2){r^2}y = (a + 2)4y$

Now, $f'(x) = 2x{a_0} + {a_1}$

$\therefore$ $f'(0) = {a_1} = 1$

and $f'(1) = 2{a_0} + {a_1} = 0$

$\Rightarrow 2{a_0} + 1 = 0$

$\Rightarrow {a_0} = - {1 \over 2}$

$\therefore$ $ay = - {1 \over 2}$

and $(a + 1)2y = 1$

$\Rightarrow 2ay + 2y = 1$

$\Rightarrow 2 \times \left( { - {1 \over 2}} \right) + 2y = 1$

$\Rightarrow 2y = + \,2$

$\Rightarrow y = + \,1$

$\therefore$ $a = - {1 \over 2}$

$\therefore$ ${a_2} = (a + 2)4y$

$= \left( { - {1 \over 2} + 2} \right) \times 4\,.\,1$

$= 6$

$\therefore$ $f(x) = - {1 \over 2}{x^2} + x + 6$

$\therefore$ $f(4) = - {1 \over 2}{(4)^2} + 4 + 6$

$= - 8 + 10$

$= 2$

"

Let for $f(x) = {a_0}{x^2} + {a_1}x + {a_2},\,f'(0) = 1$ and $f'(1) = 0$. If a₀, a₁, a₂ are in an arithmatico-geometric progression, whose corresponding A.P. has common difference 1 and corresponding G.P. has common ratio 2, then f(4) is equal to _____________.

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Let for $f(x) = {a_0}{x^2} + {a_1}x + {a_2},\,f'(0) = 1$ and $f'(1) = 0$. If a0, a1, a2 are in an arithmatico-geometric progression, whose corresponding A.P. has common difference 1 and corresponding G.P. has common ratio 2, then f(4) is equal to _____________.

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Let for $f(x) = {a_0}{x^2} + {a_1}x + {a_2},\,f'(0) = 1$ and $f'(1) = 0$. If a₀, a₁, a₂ are in an arithmatico-geometric progression, whose corresponding A.P. has common difference 1 and corresponding G.P. has common ratio 2, then f(4) is equal to _____________.