How do you determine all values of c that satisfy the mean value theorem on the interval [1,1] for #f(x) = 3x^5+5x^3+15x #?
To determine all values of ( c ) that satisfy the Mean Value Theorem on the interval ([1, 1]) for ( f(x) = 3x^5 + 5x^3 + 15x ), you need to follow these steps:
 Find the derivative of ( f(x) ), denoted as ( f'(x) ).
 Calculate the average rate of change of ( f(x) ) over the interval ([1, 1]), which is given by: [ \frac{f(1)  f(1)}{1  (1)} ]
 Find all values of ( c ) such that ( f'(c) ) equals the average rate of change calculated in step 2.
So, here are the steps more specifically:

Find the derivative: [ f'(x) = 15x^4 + 15x^2 + 15 ]

Calculate the average rate of change: [ \text{Average rate of change} = \frac{f(1)  f(1)}{1  (1)} ] [ = \frac{f(1)  f(1)}{2} ]
Now, calculate ( f(1) ) and ( f(1) ): [ f(1) = 3(1)^5 + 5(1)^3 + 15(1) = 3 + 5 + 15 = 23 ] [ f(1) = 3(1)^5 + 5(1)^3 + 15(1) = 3  5  15 = 23 ]
Substitute these values into the formula for the average rate of change: [ \text{Average rate of change} = \frac{23  (23)}{2} = \frac{46}{2} = 23 ]
 Find ( c ) such that ( f'(c) = 23 ): [ 15c^4 + 15c^2 + 15 = 23 ]
Solve this equation to find all possible values of ( c ).
By signing up, you agree to our Terms of Service and Privacy Policy
There are two values
We take only the positive root
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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)=(x^2e^x)/(x2)# increasing or decreasing at #x=1#?
 How do you find the local extrema for #f(x) = 22x^2# on domain #1 <= x <= 1#?
 How do you find the local max and min for #f (x) = x^(3)  6x^(2) + 5#?
 How do you find the critical numbers of #y= 2xtanx#?
 Is #f(x)=1/x1/x^3+1/x^5# increasing or decreasing at #x=2#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7