Question

If $f\left(0\right)=3,f\left(2\right)=5$, and $f$ is linear then, find the value of ${\int }_0^{1}x{f}^{\prime }\left(2x\right)dx.$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (B)

$1$

Let $f\left(x\right)=ax+b$

$f\left(0\right)=3\to b=3$

$f\left(2\right)=5\to 5=2a+b$ gives $a=1$

$f\left(x\right)=x+3$

$={\int }_0^{1}x\frac{d(2x+6)}{dx}dx$

$={\int }_0^{1}2xdx$

$=[x^2{]}_0^1$

$=1$ $B$ is correct