For what range of constants, will function increase everywhere

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This calculus problem from the fourth edition of James Stewart Concepts  and Contexts asks to find a range of values for the constant c such that the function is increasing on the entire real number line. A function is increasing when its first derivative is positive. So we set f prime of x greater than zero.

This leads to c being strictly greater than a quotient whose maximum value is 1/8. This can be shown by taking the derivative of the quotient to determine where the quotient has horizontal tangent lines. Horizontal tangent lines occur when the first derivative is equal to zero. As mentioned, the original function will be increasing everywhere when the constant c lies in the interval 1/8 to infinity.

Problem 72 from Calculus fourth edition by James Stewart
Problem 72 from Calculus fourth edition by James Stewart
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