Question

The curvature function of some function is given to be k(x) = \(\frac{1}{[2+2x+x^2]^{\frac{3}{2}}}\) then which of the following functions could be f(x)?

(a) ^x^2⁄2 + x + 101

(b) ^x^2⁄4 + 2x + 100

(c) x^2 + 13x + 101

(d) x^3 + 4x^2 + 1019

The question was posed to me during an interview.

My query is from Curvature in portion Differential Calculus of Engineering Mathematics

Answer 1

The correct option is (a) ^x^2⁄2 + x + 101

Best explanation: The equation for curvature is

k(x) = \(\left |\frac{f”(x)}{(1+[f'(X)]^2)^{\frac{3}{2}}} \right |\)

Observe that there is no   term in the numerator of given curvature function. Hence, the function has to be of the quadratic form. Using  f(x) = ax^2+bx+c we have

k=\(\left |\frac{2a}{[1+(2a+b)^2]^{\frac{3}{2}}} \right |=\left |\frac{1}{[2+2x+x^2]^{\frac{3}{2}}} \right |\)

\(\left |\frac{2a}{[1+(2ax+b)^2]^{\frac{3}{2}}} \right |=\left |\frac{1}{[1+(x+1)^2]^{\frac{3}{2}}} \right |\)

2a = 1; b = 1

The values of a, b  are

a = ^1⁄2; b = 1

c  Could be anything.