Question

answer the following true or false: if ( f(x) ) is a differentiable function such that ( f^{prime}(7)=0 ), then ( f(x) ) either has a relative minimum at ( x = 7 ) or a relative maximum at ( x = 7 ). true false

Explanation:

Step1: Recall the critical point concept

A critical point of a function \(y = f(x)\) is a point \(x = c\) where \(f^{\prime}(c)=0\) or \(f^{\prime}(c)\) does not exist.

Step2: Analyze the nature of critical points

Just because \(f^{\prime}(7) = 0\) (so \(x = 7\) is a critical point), it is not necessarily a relative minimum or maximum. For example, consider the function \(y=x^{3}\). The derivative \(y^{\prime}=3x^{2}\), and \(y^{\prime}(0)=0\). But for \(y = x^{3}\), the function is increasing for \(x>0\) (\(y^{\prime}=3x^{2}>0\) when \(x>0\)) and increasing for \(x < 0\) (\(y^{\prime}=3x^{2}>0\) when \(x<0\)). So \(x = 0\) is a critical point (since \(y^{\prime}(0) = 0\)) but it is not a relative maximum or minimum.

Answer:

False