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JEE Main · 2024 · 27 Jan 2024 · Shift 1 · Differentiation

Let

f (x) = x^{3} + x^{2} f^{'} (1) + x f^{''} (2) + f^{'''} (3)

,

x \in R

. Then

f^{'} (10)

is equal to ____.

Answer: 202

Solution

Differentiating

f (x) = x^{3} + x^{2} f^{'} (1) + x f^{''} (2) + f^{'''} (3)

repeatedly (the last term is a constant):

f^{'} (x) = 3 x^{2} + 2 x f^{'} (1) + f^{''} (2), f^{''} (x) = 6 x + 2 f^{'} (1), f^{'''} (x) = 6

Let

A = f^{'} (1), B = f^{''} (2)

. Then

B = 12 + 2 A

, and evaluating

f^{'}

at

x = 1

:

A = 3 + 2 A + B = 3 + 2 A + 12 + 2 A

, giving

A = - 5

, so

B = 2

.

f^{'} (x) = 3 x^{2} - 10 x + 2 \Rightarrow f^{'} (10) = 300 - 100 + 2 = 202

JEE Main · 2024 · 27 Jan 2024 · Shift 1 · Integer answer

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