Let f x and g x be twice differentiable functions on 0 2 satisfying f 39 39 x g 39 39 x f 39 1 4 g 3...........

Mathematics (NDA) Applications of Derivatives

Let f (x ) and g(x ) be twice differentiable functions on [0, 2] satisfying f ''(x ) = g'' (x ), f'(1) = 4, g'(1) = 6, f (2) = 3 and g(2) = 9. Then, what is f (x ) − g(x ) at x = 4 equal to?

-10

-6

-4

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Explanation:

Given, f''(x) = g''(x)
On integrating both sides, we get
f '(x) = g'(x) + c
=> f'(1) = g'(1) + c
=> 4 = 6 + c
=> c = − 2
$\therefore$ f'(x) = g'(x) − 2
Again, on integrating both sides, we get
f (x) = g(x) − 2x + c₁
=> f (2) =g(2)- 2 x 2 + c₁
=> 3= 9 -4 + c₁
=> c₁ = − 2
$\therefore$ f (x) − g(x) = − 2x − 2
At x = 4, [ f (x) − g(x)] = − 8 − 2 = − 10