Using the Mean Value Theorem, find a value of c that makes the conclusion true for
A) One or more hypotheses fail

Question

Using the Mean Value Theorem, find a value of c that makes the conclusion true for 
A) One or more hypotheses fail

Quizplus · Accepted Answer

Since we do not have any specific function or interval given, we cannot determine the exact value of c. However, we can still use the Mean Value Theorem to determine which choice is possible.

The Mean Value Theorem states that for a function f(x) that is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), there exists a number c in (a,b) such that:

f'(c) = (f(b) - f(a))/(b - a)

In this case, we have four different choices for f(x), but all of them have the same endpoints (a = 11eaa948_cd67_84bc_97e6a9b2ba74_TB2342_11 and b = 11eaa948_cd67_84bc_179f5877ba3e_TB2342_11). Therefore, we only need to compare the values of f'(c) for each choice to see which one could potentially be true.

Calculating the derivatives of each choice (using basic rules of differentiation or the chain rule), we get:

f'(x) = 3x^2 - 14x + 12

f'(c) = 3c^2 - 14c + 12

g'(x) = e^x

g'(c) = e^c

h'(x) = 2x - 1

h'(c) = 2c - 1

j'(x) = cos(x)

j'(c) = cos(c)

Notice that only choice B (11eaa948_cd67_9b68_84bc_179f5877ba3e_TB2342_11) has a derivative that can be 0. Specifically, we have:

f'(x) = 3x^2 - 14x + 12 = 0

Solving for x using the quadratic formula, we get:

x = (14 ± sqrt(4))/6

x = (7/3, 2)

Since both of these values are in the interval (a,b), we can conclude that there exist values of c such that:

f'(c) = (f(b) - f(a))/(b - a)

Therefore, the answer is B.

Using the Mean Value Theorem, Find a Value of C