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JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Differentiation

If

f (x) = x^{2} + g^{'} (1) x + g^{''} (2)

and

g (x) = f (1) x^{2} + x f^{'} (x) + f^{''} (x)

,
then the value of

f (4) - g (4)

is equal to ____.

Answer: 14

Solution

Since

f (x) = x^{2} + g^{'} (1) x + g^{''} (2)

, we have

f^{'} (x) = 2 x + g^{'} (1)

and

f^{''} (x) = 2

. Substituting these into

g (x) = f (1) x^{2} + x f^{'} (x) + f^{''} (x)

gives

g (x) = (f (1) + 2) x^{2} + g^{'} (1) x + 2

, so

g^{''} (x) = 2 (f (1) + 2)

and

g^{'} (x) = 2 (f (1) + 2) x + g^{'} (1)

. Matching

g^{'} (1) = 2 (f (1) + 2) + g^{'} (1)

forces

f (1) = - 2

, and then

g^{''} (2) = 2 (f (1) + 2) = 0

. From

f (1) = 1 + g^{'} (1) + g^{''} (2) = - 2

it follows that

g^{'} (1) = - 3

, so

f (x) = x^{2} - 3 x

and

g (x) = - 3 x + 2

. Therefore

f (4) - g (4) = (16 - 12) - (- 12 + 2) = 14

.

JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Integer answer

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