Question

Sketch the graph of the function g for which \(g\left( {\bf{0}} \right) = g\left( {\bf{2}} \right) = g\left( {\bf{4}} \right) = {\bf{0}}\), \(g'\left( {\bf{1}} \right) = g'\left( {\bf{3}} \right) = {\bf{0}}\), \(g'\left( {\bf{0}} \right) = g'\left( {\bf{4}} \right) = {\bf{1}}\), \(g'\left( {\bf{2}} \right) = - {\bf{1}}\), \(\mathop {{\bf{lim}}}\limits_{x \to \infty } g\left( x \right) = \infty \), and \(\mathop {{\bf{lim}}}\limits_{x \to - \infty } g\left( x \right) = - \infty \).

Short answer

The sketch of the graph of the function is shown below:

Step-by-step solution

Step 1:Write an interpretation of given information for g

Following observations can be made about the function \(g\left( x \right)\):

1. The equation \(g\left( 0 \right) = g\left( 2 \right) = g\left( 4 \right) = 0\) shows the curve of g is intersecting the x-axis at \(x = 0\), \(x = 2\) and \(x = 4\).
2. The equation \(g'\left( 1 \right) = g'\left( 3 \right) = 0\) shows that at \(x = 1\) and \(x = 3\), the tangent slope is zero, i.e., the tangent is parallel to thex-axis.
3. It can be observed from the equation \(g'\left( 2 \right) = - 1\), the slope at \(x = 2\) is \( - 1\).
4. The equation \(\mathop {\lim }\limits_{x \to \infty } g\left( x \right) = \infty \) shows as x is approaching to \(\infty \), g is also approaching to \(\infty \).
5. The equation \(\mathop {\lim }\limits_{x \to - \infty } g\left( x \right) = - \infty \) shows as x is approaching to \(\infty \), g is also approaching to \(\infty \).

Sketch the graph of the function

The figure below represents the graph of \(g\left( x \right)\).

Thus, the sketch of the curve function is obtained.