Question-4

Find the global minima of the function \(f(x) = 2x^3 - 9x^2 + 12x + 6\) in the interval \((1, 3)\). You need to enter the value of \(x^{*}\) and not the value of \(f(x^{*})\).

Answer

\(2\)

Solution

We have \(f^{\prime}(x) = 6x^2 - 18x + 12 = 6(x - 1)(x - 2)\) and \(f^{\prime \prime}(x) = 6(2x - 3)\). The point \(x = 1\) corresponds to a local maxima and \(x = 2\) is a local minima. \(f(1) = 11\) and \(f(2) = 10\) and \(f(3) = 15\). The function decreases from \(x = 1\) to \(x = 2\), attains a minima at \(x = 2\) and then increases from \(x = 2\) to \(x = 3\). From this, we can conclude that \(x = 2\) is the global minima of \(f\) in the open interval \((1, 3)\).