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Question

If $$f(x)$$ and $$g(x)$$ are twice differentiable functions on $$(0, 3)$$ satisfying $$f''(x) = g'', f'(1) = 4, g'(1) = 6, f(2) = 3, g(2) = 9$$, then $$f(1) - g(1)$$ is:


A
4
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B
4
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C
0
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D
2
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Solution

The correct option is B $$-4$$
According to question,

$$f''(x) = g''(x)$$

Integrating w.r.t. $$x$$, we get

$$f'(x) = g'(x) + C_{1}$$

Put $$x = 1\Rightarrow f'(1) = g'(1) + C_{1}$$

$$\Rightarrow 4 = 6 + C_{1}$$

$$\therefore C_{1} = 2$$

$$\therefore f'(x) = g'(x) - 2$$

Again, integrating w.r.t. $$x$$, we get

$$f(x) = g(x) - 2x + C_{2}$$

$$\Rightarrow 3 = 9 - 4 + C_{2} \Rightarrow C_{2} = -2$$

$$\therefore f(x) = g(x) - 2x - 2$$

Put $$x = 1$$, we get

$$f(1) - g(1) = -2(1) - 2 = -4$$

Mathematics

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