Question

Suppose $g\left(x\right)=2x+1$ and $h\left(x\right)=4x^2+4x+5$ and $h\left(x\right)=\left(fog\right)\left(x\right)$ The area enclosed by the graph of the function $y=f\left(x\right)$ and the pair of tangents drawn to it from the origin is

• A. 8/3
• B. 16/3
• C. 32/3
• D. none

Answer 1

The correct option is B 16/3

$g\left(x\right)=2x+1$ and
$h\left(x\right)=4x^2+4x+5=(2x+1{)}^2+4=g(x{)}^2+4$
$h\left(x\right)=\left(fog\right)\left(x\right)$
$f\left(x\right)=x^2+4$
Let the point where tangent touches the parabola be (x,y)= $(x,x^2+4)$
Slope = $2x$
Equation of tangent
$\frac{x^2+4}{x}=\frac{2x}{1}$
$x=\pm 2$
$y=8$
Area enclosed
$=2(\frac{1}{2}4\times 8-{\int }_4^8\sqrt{y-4}dy)$
$=\frac{16}{3}$