Question

Find an equation of a parabola that has curvature 4 at the origin.

Short answer

The required equation of the parabolas is \(f(x) = 2{x^2}\)and \(f(x) = - 2{x^2}\)s

Step-by-step solution

Step 1: Calculate the curvature using the formula

Let us assume an equation of the form\(f(x) = a{x^2},a \ne 0.\)

\(k = \frac{{\left| {{f^{\prime \prime }}(x)} \right|}}{{{{\left( {1 + {{\left( {{f^\prime }(x)} \right)}^2}} \right)}^{\frac{3}{2}}}}}\)

\({f^\prime }(x) = 2ax\)

\({f^{\prime \prime }}(x) = 2a\)

\(k = \frac{{|2a|}}{{{{\left( {1 + {{(2ax)}^2}} \right)}^{\frac{3}{2}}}}}\)

\(k = \frac{{2|a|}}{{{{\left( {1 + 4{a^2}{x^2}} \right)}^{\frac{3}{2}}}}}\)

Step 2: Evaluate the curvature at the origin

Assess the curvature at the origin now and set\(k(0) = 4\)

\(k(0) = 2|a| = 4\)

\(a = 2\)or\(a = - 2\)

Thus, the equation of the parabola is

\(f(x) = 2{x^2}\) or \(f(x) = - 2{x^2}\)