Science:Math Exam Resources/Courses/MATH100/December 2015/Question 05 (b)/Solution 1

The line tangent to the curve $y=f(x)$ at $x=3$ has slope 2 means $f'(3)=2$.

Similarly, since the line tangent to the curve $y=g(x)$ at $x=3$ has slope 5, we have $g'(3)=5$.

Then, using the product rule with $g(x)=x\cdot f(x)$,

$g'(x)=f(x)+x\cdot f'(x)$ , hence at $x=3$ we have

$5=g'(3)=f(3)+3\cdot f'(3)=f(3)+6$.

Therefore, the answer is $f(3)=-1$.