Given the function #f(x)=x/(x+9)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and find the c?
Please see below.
The Mean Value Theorem has two Hypotheses. A hypothesis is satisfied if it is true.
By signing up, you agree to our Terms of Service and Privacy Policy
To determine if the function f(x) satisfies the hypotheses of the Mean Value Theorem on the interval [1,4], we need to check if two conditions are met:
- The function f(x) must be continuous on the closed interval [1,4].
- The function f(x) must be differentiable on the open interval (1,4).
Firstly, f(x) = x/(x+9) is continuous for all real numbers except where the denominator is zero. Since x+9 ≠ 0 for all real numbers, f(x) is continuous everywhere. Therefore, it is continuous on the closed interval [1,4].
Next, to check differentiability, we need to ensure that the derivative of f(x) exists for all x in the open interval (1,4). The derivative of f(x) can be found using the quotient rule:
f'(x) = (1*(x+9) - x*(1))/(x+9)^2 = 9/(x+9)^2
This derivative is defined for all x ≠ -9, which is true for all x in the interval (1,4).
Since both conditions are met, f(x) satisfies the hypotheses of the Mean Value Theorem on the interval [1,4].
To find the c value guaranteed by the Mean Value Theorem, we use the formula:
f'(c) = (f(4) - f(1))/(4 - 1)
Plugging in the values:
f'(c) = (4/(4+9) - 1/(1+9))/(4 - 1) = (4/13 - 1/10)/3
Solving this equation will give us the value of c.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
- Is #f(x)=sin(pi/2-x)-cos(pi-x)# increasing or decreasing at #x=pi/3#?
- How do you use the mean value theorem to prove that #sinx-siny = x-y#?
- How do you find the local extrema for #f(x) = (x-3)^3# on (-∞, ∞)?
- How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for # f (x) = x + 4/x #?
- Is #f(x)=x/sqrt(x+3) # increasing or decreasing at #x=5 #?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7