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

Step 1: Simplify the given function. Divide the numerator by the denominator: \[ \frac{x^4+x^2+1}{x^2+x+1} = x^2-x+1-\frac{x}{x^2+x+1}. \] Differentiating, \[ y_1 = 2x-1 - \frac{(x^2+x+1)-x(2x+1)} {(x^2+x+1)^2}. \]

Step 2: Find the second derivative. Differentiating once again, \[ y_2 = 2+\frac{2x^3+3x^2-3x-1}{(x^2+x+1)^3}. \] Now substitute \[ x=\frac12+\sqrt2. \] This gives \[ y_1=2\sqrt2, \] and \[ y_2=\frac{16\sqrt2}{27}. \]

Step 3: Evaluate the required expression. Now, \[ 1+y_1^2 = 1+8 = 9. \] Hence, \[ (1+y_1^2)^{3/2} = 9^{3/2} = 27. \] Therefore, \[ \left| \frac{(1+y_1^2)^{3/2}}{y_2} \right| = \frac{27}{2}. \] Hence, \[ \boxed{\frac{27}{2}}. \] Thus, \[ \boxed{(C)} \] is the correct answer.