For the differentiable function f:R-0 R, let 3f(x)+2f((1)/(x))=(1)/(x)-10, then |f(3)+f^'(…

Question

For the differentiable function $f:R-{0}	o R$, let $3f(x)+2f(\frac{1}{x})=\frac{1}{x}-10$, then $|f(3)+{f}^{'}(\frac{1}{4})|$ is equal to

Accepted Answer

Given the equation: $3f(x)+2f(\frac{1}{x})=\frac{1}{x}-10$

Replace $x$ with $\frac{1}{x}$ in the original equation:

$3f(\frac{1}{x})+2f(x)=x-10$

Now, we have two equations:

$3f(x)+2f(\frac{1}{x})=\frac{1}{x}-10$

$3f(\frac{1}{x})+2f(x)=x-10$

By adding the two equations, we can find $f(x)$:

$5f(x)=\frac{3}{x}-2x-10$

Now, let's differentiate both sides with respect to $x$:

$5{f}^{'}(x)=-\frac{3}{{x}^{2}}-2$

Now, we can find the values for $f(3)$ and ${f}^{'}(\frac{1}{4})$:

$f(3)=\frac{1}{5}(1-6-10)=-3$

${f}^{'}(\frac{1}{4})=\frac{1}{5}(-48-2)=-10$

Finally, calculate the expression we are interested in :

$|f(3)+{f}^{'}(\frac{1}{4})|=|-3-10|=13$

Solutions · 1