Let f(x) and g(x) be twice differentiable functions satisfying f^''(x)=g^''(x) for all x R…

Question

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying ${f}^{''}(x)={g}^{''}(x)$ for all $x\in R,{f}^{'}(1)=2{g}^{'}(1)=4$ and $g(2)=3f(2)=9$. Then $f(25)-g(25)$ is equal to :

Accepted Answer

Let us define a new function $h(x)$ taking the difference between $f(x)$ and $g(x)$.

$h(x)=f(x)-g(x)$

Differentiating this function twice with respect to $x$, we get:

${h}^{''}(x)={f}^{''}(x)-{g}^{''}(x)$

We are given that ${f}^{''}(x)={g}^{''}(x)$ for all $x\in R$. Therefore, their difference is zero:

${h}^{''}(x)=0$

Now, let us integrate both sides with respect to $x$ to find ${h}^{'}(x)$:

${h}^{'}(x)=\int 0\, dx={c}_{1}$

where ${c}_{1}$ is a constant of integration.

We are given the values of the first derivatives at $x=1$. Let us simplify them:

${f}^{'}(1)=4$

$2{g}^{'}(1)=4\, \implies \, {g}^{'}(1)=2$

Using these values, we can find the exact value of ${h}^{'}(1)$:

${h}^{'}(1)={f}^{'}(1)-{g}^{'}(1)=4-2=2$

Since ${h}^{'}(x)={c}_{1}$ for all values of $x$, it follows that:

${c}_{1}=2$

Substituting this back, we get our first derivative equation:

${h}^{'}(x)=2$

Next, we need the original function $h(x)$. We find it by integrating ${h}^{'}(x)$ with respect to $x$:

$h(x)=\int 2\, dx=2x+{c}_{2}$

where ${c}_{2}$ is another constant of integration.

We are given the values of the functions at $x=2$. Let us simplify these values as well:

$g(2)=9$

$3f(2)=9\, \implies \, f(2)=3$

Using these values, we can find the value of $h(2)$:

$h(2)=f(2)-g(2)=3-9=-6$

Now, substitute $x=2$ and $h(2)=-6$ into our integrated equation for $h(x)$:

$h(2)=2(2)+{c}_{2}$

$-6=4+{c}_{2}$

${c}_{2}=-6-4$

${c}_{2}=-10$

Substituting the value of ${c}_{2}$ back into the equation for $h(x)$, we have the completed function rule:

$h(x)=2x-10$

The question asks for the value of $f(25)-g(25)$, which is represented directly by $h(25)$. Let us substitute $x=25$ into our function $h(x)$:

$h(25)=2(25)-10$

$h(25)=50-10$

$h(25)=40$

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

The correct answer is Option B.

Solutions · 1