Question

let ( g ) be a differentiable function such that ( g(3)=2 ) and ( g^{prime}(3)=-\frac{3}{4} ). the graph of ( g ) is concave down on the interval ( (2,4) ). which of the following is true about the approximation for ( g(2.6) ) found using the line tangent to the graph of ( g ) at ( x = 3 )?

a ( g(2.6) approx 1.7 ) and this approximation is an overestimate of the value of ( g(2.6) )
b ( g(2.6) approx 1.7 ) and this approximation is an underestimate of the value of ( g(2.6) )
c ( g(2.6) approx 2.3 ) and this approximation is an overestimate of the value of ( g(2.6) )
d ( g(2.6) approx 2.3 ) and this approximation is an underestimate of the value of ( g(2.6) )

Explanation:

Step1: Write the tangent line equation

The equation of the tangent line to \(y = g(x)\) at \(x = a\) is \(L(x)=g(a)+g^{\prime}(a)(x - a)\). Here \(a = 3\), \(g(3)=2\) and \(g^{\prime}(3)=-\frac{3}{4}\). So \(L(x)=2-\frac{3}{4}(x - 3)\).

Step2: Approximate \(g(2.6)\)

Substitute \(x = 2.6\) into \(L(x)\):

$$
LATEXBLOCK0
$$

Step3: Analyze the concavity

Since the graph of \(g\) is concave - down on the interval \((2,4)\), the function \(g(x)\) lies below its tangent line on the interval \((2,4)\) (by the property of concave - down functions: \(g(x)\leq L(x)\) for \(x\) in the interval where the function is concave - down and \(L(x)\) is the tangent line).

Answer:

A. \(g(2.6)\approx1.7\) and this approximation is an overestimate of the value of \(g(2.6)\)