Question

Identify a function \(f\) that has the following characteristics. Then sketch the function. \(f(0)=4 ; f^{\prime}(0)=0\) \(f^{\prime}(x)<0\) for \(x<0\) \(f^{\prime}(x)>0\) for \(x>0\)

Short answer

The function that meets the given properties is \(f(x) = x^2 + 4\). The graph of this function is a parabola shifted upward by 4 units.

Step-by-step solution

Finding the function

Given the properties, we know that this function changes its behavior at x=0, where it has a local minimum or maximum or an inflection point. A logical choice matching these properties would be a parabolic function. We know the standard form of a parabola is \(f(x) = ax^2 + c\). Here, a should be positive, because the positive x squared term makes the function increase for \(x>0\) and decrease for \(x<0\). Thus, taking a=1, the equation of the function would be \(f(x) = x^2 + c\). To find c, we substitute \(x=0\) into the equation and use the fact that \(f(0)=4\) to get \(c=4\). So, the function that matches the given properties is \(f(x) = x^2 + 4\).

Sketching the function

Since the equation of the function is now known, which is \(f(x) = x^2 + 4\), it is a vertically shifted upward parabola. Start by sketching a basic parabola shape that opens upwards. Move the vertex of the parabola to the point (0, 4) on the y-axis because this is where our function intersects the y-axis. Make sure that the tangent line at the vertex is horizontal reflecting the fact that the derivative at this point is zero. The resulting sketch should show a parabolic curve that opens upwards, has the vertex at the point (0,4), and is increasing for \(x>0\) and decreasing for \(x<0\).