Chapter 1.4, Problem 41E

To determine

To find: The truth value of the given sentence.

Answer to Problem 41E

The given statement is true. The normal line is perpendicular to the tangent line at the point.

Explanation of Solution

Given information:

The graph of a function has a tangent line at $x=a$ , then the graph also has a normal line at $x=a$ .

Justification:

If the function $f\left(x\right)$ has a tangent line at $x=a$ then the equation of the tangent line is:

  $y-f\left(a\right)=f^{\prime }\left(a\right)\left(x-a\right)$ .

Slope of the normal line at $x=a$ is given by: $-\frac{1}{f^{\prime }\left(a\right)}$ .

The equation of normal line at $x=a$ is given by: $y-f\left(a\right)=-\frac{1}{f^{\prime }\left(a\right)}\left(x-a\right)$ .

Hence the given statement is true. The graph of a function has a tangent line at $x=a$ , then the graph also has a normal line at $x=a$ .

The normal line is perpendicular to the tangent line at the point.