question 1\nconsider the function ( f(x)=x^{2}-3x + 5 ). what is a value of ( -1lt clt3 ) such that\n(…

question 1\nconsider the function ( f(x)=x^{2}-3x + 5 ). what is a value of ( -1lt clt3 ) such that\n( \frac{f(3)-f(-1)}{3-(-1)}=f^{prime}(c) )?\nenter your answer below (write dne if there is no value of ( c )).

question 1\nconsider the function ( f(x)=x^{2}-3x + 5 ). what is a value of ( -1lt clt3 ) such that\n( \frac{f(3)-f(-1)}{3-(-1)}=f^{prime}(c) )?\nenter your answer below (write dne if there is no value of ( c )).

Answer

Explanation:

Step1: Calculate (f(3)) and (f(-1))

Substitute (x = 3) into (f(x)=x^{2}-3x + 5): (f(3)=3^{2}-3\times3 + 5=9 - 9+5 = 5) Substitute (x=-1) into (f(x)=x^{2}-3x + 5): (f(-1)=(-1)^{2}-3\times(-1)+5=1 + 3+5=9) Then (\frac{f(3)-f(-1)}{3-(-1)}=\frac{5 - 9}{4}=\frac{-4}{4}=-1)

Step2: Find the derivative (f^{\prime}(x))

Using the power rule ((x^{n})^\prime=nx^{n - 1}), for (f(x)=x^{2}-3x + 5), (f^{\prime}(x)=2x-3)

Step3: Solve for (c)

Set (f^{\prime}(c)=-1), so (2c-3=-1) Add (3) to both sides: (2c=-1 + 3=2) Divide both sides by (2): (c = 1)

Answer:

(1)