Find the values of \(x\), such that \(f(x)\) is increasing function\(f(x)=2x^3-15x^2-144x-7\)Solution:Given: \(f(x)=2x^3-15x^2-144x-7\)\(\therefore f'(x)=6x^2-30x-144\)Now, \(f'(x)\gt 0\), as \(f(x)\) is increasing\(\therefore 6x^2-30x-144\gt 0\)\(\t…

Q.3 (C) (i) Find the values of \(x\), such that \(f(x)\) is increasing function
\(f(x)=2x^3-15x^2-144x-7\)

Solution:
Given: \(f(x)=2x^3-15x^2-144x-7\)
\(\therefore f'(x)=6x^2-30x-144\)
Now, \(f'(x)\gt 0\), as \(f(x)\) is increasing
\(\therefore 6x^2-30x-144\gt 0\)
\(\therefore x^2-5x-24\gt 0\)
\(\therefore (x-8)(x+3)\gt 0\)
Case (I) \(x-8\gt 0\) and \(x+3\gt 0\)
\(x\gt 8\) and \(x\gt -3\)
\(\therefore x\gt\)[ ]
Case (II) \(x-8\lt 0\) and \(x+3\lt 0\)
\(x\lt 8\) and \(x\lt -3\)
\(\therefore x\lt\)[ ]
\(\therefore f(x)=2x^3-15x^2-144x-7\) is increasing if and only if \(x\in(-\infty,\)[ ]\()\) or \(x\in(\)[ ]\(,\infty)\)