Question

Identify a function \(f\) that has the following characteristics. Then sketch the function. \(f(0)=0 ; f^{\prime}(0)=0 ; f^{\prime}(x)>0\) if \(x \neq 0\)

Short answer

The function that satisfies the given conditions is \(f(x) = x^2\). The sketch would be an upward facing parabola, originating from the origin and increasing on either side.

Step-by-step solution

Interpret the function characteristics

Here, we interpret that the function \(f\) passes through the origin (given by \(f(0) = 0\)), it has a horizontal tangent at the origin (given by \(f^{\prime}(0) = 0\)), and the function is increasing to either side of the origin (given by \(f^{\prime}(x) > 0\) for \(x \neq 0\)). This suggests a function which 'bottoms out' or 'tops out' at the origin and increases on either side. This sounds like a parabola, and based on the 'increasing' part, it would have to open upwards.

Identify a suitable function

Since it appears to be a parabola, we can guess that the function is of the form \(f(x) = ax^2\). We know that \(f(0) = 0\), so the parabola must pass through the origin. So it is reasonable to consider that \(a\) has a positive value, as the function is increasing to either side of the origin. Therefore, a simple function in form of \(f(x) = x^2\) can be identified as satisfying all the conditions.

Sketch the function

Draw an upward-facing parabola with its vertex at the origin. The parabola will be increasing to the right of the origin and to the left as well. Initiate from the vertex at the origin and draw the graph in such a way that it keeps increasing in positive direction on either side of the origin.