Question

in the figure, the straight line is tangent to the graph of ( f(x) ). find ( f(5) ) and ( f^{prime}(5) ).

( f(5)= )
(type an integer or a fraction. simplify your answer.)

( f^{prime}(5)= )
(type an integer or a fraction. simplify your answer.)

Explanation:

Step1: Find \( f(5) \)

Since the tangent line touches the graph of \( y = f(x) \) at \( x = 5 \) and the \( y \)-coordinate of the point of tangency is given as \( 7 \), by the definition of a function \( y=f(x) \), when \( x = 5 \), \( f(5)=7 \).

Step2: Find the equation of the tangent line

The general form of a line is \( y=mx + b \). We know that the line passes through the points \( (0,1) \) and \( (5,7) \).
The slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1,y_1)=(0,1) \) and \( (x_2,y_2)=(5,7) \).
\( m=\frac{7 - 1}{5-0}=\frac{6}{5} \).
The derivative of the function \( f(x) \) at \( x = 5 \), \( f^{\prime}(5) \), is equal to the slope of the tangent line at \( x = 5 \).

Answer:

\( f(5)=7 \)
\( f^{\prime}(5)=\frac{6}{5} \)