Question

If \[ y=\frac{x^4+x^2+1}{x^2+x+1}, \] \[ \frac{dy}{dx}=y_1 \] and \[ \frac{d^2y}{dx^2}=y_2, \] then the value of \[ \left| \frac{(1+y_1^2)^{3/2}}{y_2} \right| \] at \[ x=\left(\frac12+\sqrt2\right) \] is

• A. \(12\sqrt2\)
• B. \(\dfrac{27}{\sqrt2}\)
• C. \(\dfrac{27}{2}\)
• D. \(\dfrac{30\sqrt2}{7}\)

Hint:

The expression \[ \boxed{ \frac{(1+y'^2)^{3/2}}{|y''|} } \] represents the radius of curvature. Simplify the function first before differentiating to reduce computation.

Answer 1

The Correct Option is C