Question

Sketch a graph of a differentiable function \(f\) that satisfies the following conditions and has \(x=2\) as its only critical number. \(f^{\prime}(x)<0\) for \(x<2 \quad f^{\prime}(x)>0\) for \(x>2\) \(\lim _{x \rightarrow-\infty} f(x)=\lim _{x \rightarrow \infty} f(x)=6\)

Short answer

The graph of function \(f\) has a minimum at \(x=2\) and a horizontal asymptote at \(y=6\). To the left of \(x=2\), the graph is decreasing and to the right, the graph is increasing. Both sides trend towards \(y=6\) as \(x\) approaches infinity and negative infinity.

Step-by-step solution

Identify the Impact of given Derivative Conditions

The derivative's sign indicates the function's direction. If \(f^{\prime}(x)<0\), the function is decreasing and if \(f^{\prime}(x)>0\), the function is increasing. Hence, since \(f^{\prime}(x)<0\) for \(x<2\) and \(f^{\prime}(x)>0\) for \(x>2\), this implies that \(f\) is decreasing until \(x = 2\) and then starts increasing. Thus, the function has a minimum at \(x =2\).

Use the Limit to Identify the Asymptote

Given that \(\lim _{x\rightarrow-\infty} f(x)=\lim _{x \rightarrow \infty} f(x)=6\), it is known that the function approaches the value 6 as \(x\) tends towards infinity or negative infinity. This implies that the function has a horizontal asymptote at \(y=6\) as the value of \(x\) tends either towards positive infinity or negative infinity.

Sketch the Graph

Using the above informations, start by drawing a sketch of the horizontal asymptote at \(y=6\). Coming to the critical point at \(x=2\), make a point below \(y=6\) since it's a minimum. The left part of the graph should be decreasing towards the asymptote and the right part should be increasing towards the asymptote.