Question

Find the number \(c\) that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at \(\left( {c,f\left( c \right)} \right)\). Are the secant line and the tangent line parallel?

19. \(f\left( x \right) = \sqrt x ,\,\,\,\,\,\,\,\,\,\,\,\,\left( {0,4} \right)\)

Short answer

The number \(c = 1\) satisfies its conclusion of the Mean Value Theorem.

The graph obtained is:

Yes, the secant line and tangent line are parallel to each other.

Step-by-step solution

Mean Value Theorem

The Mean Value Theorem is applicable for any function \(f\left( x \right)\) if and only if

\(\begin{array}{l}f\left( x \right) \to {\rm{continuous within }}\left( {a,b} \right)\\f\left( x \right) \to {\rm{differentiable within }}\left( {a,b} \right)\\{\rm{then, for }}c \in \left( {a,b} \right) \to f'\left( c \right) = \frac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\end{array}\).

Applying the Mean Value theorem to the function:

The given function is:

\(f\left( x \right) = \sqrt x ,\,\,\,\,\,\,\,\,\,\,\,\,\left( {0,4} \right)\)

Now, according to the Mean Value Theorem, there exists a point \(c \in \left( {0,4} \right)\)such that:

\(f'\left( x \right)\left| {_{x = c}} \right. = \frac{{f\left( 4 \right) - f\left( 0 \right)}}{{4 - 0}}\)

On solving, we have:

\(\begin{array}{c}f'\left( x \right)\left| {_{x = c}} \right. = \frac{{f\left( 4 \right) - f\left( 0 \right)}}{{4 - 0}}\\\frac{1}{{2\sqrt c }} = \frac{{2 - 0}}{4}\\\frac{1}{{2\sqrt c }} = \frac{1}{2}\\c = 1\end{array}\)

Hence, the number \(c = 1\) satisfies the conclusion.

Graph for secant and tangent lines:

The procedure to draw the graph of the above equation by using the graphing calculator is as follows:

To check the answer visually draw the graph of the function\({f_1}\left( x \right) = \sqrt x \)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(\sqrt x \)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of graph of the function is shown below:

Hence, we see the tangent line and the secant line are parallel to each other.