Difference of Differentiable Functions Given Second Derivatives Equality

To find the value of $f(25) - g(25)$, we define a new function: $h(x) = f(x) - g(x)$

Differentiating $h(x)$ twice with respect to $x$, we get: $h'(x) = f'(x) - g'(x)$ $h''(x) = f''(x) - g''(x)$

Given that $f''(x) = g''(x)$ for all $x \in \mathbb{R}$, it follows that: $h''(x) = 0$

Integrating $h''(x) = 0$ with respect to $x$ yields: $h'(x) = A$ where $A$ is a constant.

Integrating $h'(x) = A$ again gives: $h(x) = Ax + B$ where $B$ is another constant.

Now, we determine the values of the constants $A$ and $B$ using the given conditions:

1. Finding $A$: We are given $f'(1) = 4$ and $2g'(1) = 4 \implies g'(1) = 2$. $h'(1) = f'(1) - g'(1) = 4 - 2 = 2$ Since $h'(x) = A$ for all $x$, we have: $A = 2$

2. Finding $B$: We are given $g(2) = 9$ and $3f(2) = 9 \implies f(2) = 3$. $h(2) = f(2) - g(2) = 3 - 9 = -6$ Using $h(x) = 2x + B$ at $x = 2$: $h(2) = 2(2) + B = -6$ $4 + B = -6 \implies B = -10$

Thus, the function $h(x)$ is given by: $h(x) = 2x - 10$

Finally, we calculate $f(25) - g(25) = h(25)$: $h(25) = 2(25) - 10 = 50 - 10 = 40$

Therefore, $f(25) - g(25) = 40$.

Correct Option: B