let f(x)=((-1/5)(x-4)^(5/3))-(2(x-4)^(2/3)-1)). find the interval (a,b) wherer f increases and give the answer as a+b

Given the function:

\[
f(x) = \left( -\frac{1}{5}(x-4)^{\frac{5}{3}} \right) - \left( 2(x-4)^{\frac{2}{3}} - 1 \right)
\]

### Step 1: Find the derivative of \( f(x) \)

To determine where \( f(x) \) is increasing, we need to compute its derivative, \( f'(x) \), and analyze where it's positive.

#### First term:
\[
f_1(x) = -\frac{1}{5}(x-4)^{\frac{5}{3}}
\]
Using the chain rule for \( (x-4)^{\frac{5}{3}} \), we get:
\[
f_1'(x) = -\frac{1}{5} \cdot \frac{5}{3}(x-4)^{\frac{2}{3}} = -\frac{1}{3}(x-4)^{\frac{2}{3}}
\]

#### Second term:
\[
f_2(x) = -2(x-4)^{\frac{2}{3}} + 1
\]
The derivative of this term is:
\[
f_2'(x) = -2 \cdot \frac{2}{3}(x-4)^{-\frac{1}{3}} = -\frac{4}{3}(x-4)^{-\frac{1}{3}}
\]

### Step 2: Combine the derivatives
Now, combine both derivative terms:

\[
f'(x) = -\frac{1}{3}(x-4)^{\frac{2}{3}} - \frac{4}{3}(x-4)^{-\frac{1}{3}}
\]

Factor out \( \frac{1}{3}(x-4)^{-\frac{1}{3}} \):

\[
f'(x) = \frac{1}{3}(x-4)^{-\frac{1}{3}} \left( - (x-4) - 4 \right) = \frac{1}{3}(x-4)^{-\frac{1}{3}} \left( -(x-4) - 4 \right)
\]

Simplify the expression inside the parentheses:

\[
f'(x) = \frac{1}{3}(x-4)^{-\frac{1}{3}} \left( -x + 4 - 4 \right) = \frac{1}{3}(x-4)^{-\frac{1}{3}} (-x)
\]

Thus, the derivative simplifies to:

\[
f'(x) = -\frac{x}{3}(x-4)^{-\frac{1}{3}}
\]

### Step 3: Analyze the derivative
To find the intervals where \( f(x) \) is increasing, we need to solve \( f'(x) > 0 \):

\[
-\frac{x}{3}(x-4)^{-\frac{1}{3}} > 0
\]

This inequality is positive when:
1. \( x > 4 \) (so that \( (x-4)^{-\frac{1}{3}} \) is positive)
2. \( x < 0 \) to ensure the negative sign in front makes the whole expression positive.

Thus, the function is increasing for \( 0 < x < 4 \).

### Step 4: Find \( a + b \)
The interval where \( f(x) \) is increasing is \( (0, 4) \). So \( a = 0 \) and \( b = 4 \).

Thus, \( a + b = 0 + 4 = 4 \).

### Final Answer:
The value of \( a + b \) is \( \boxed{4} \).

Would you like a deeper explanation on any step?

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Here are 5 follow-up questions:
1. How do you find the critical points of a function using its derivative?
2. Why is the sign of \( f'(x) \) important when determining intervals of increase or decrease?
3. Can you explain the chain rule applied in this problem in more detail?
4. How does factoring help in simplifying derivatives for solving inequalities?
5. What role do fractional exponents play in differentiation?

**Tip:** Always double-check the domain of the original function when solving inequalities involving fractional exponents.

let f(x)=((-1/5)(x-4)^(5/3))-(2(x-4)^(2/3)-1)). find the interval (a,b) where f increases and give the answer as a+b

Solve the problem of finding the interval where f(x) = (-1/5)(x-4)^(5/3) - 2(x-4)^(2/3) + 1 is increasing. Learn to compute the derivative and apply inequality analysis to determine the increasing interval. Suitable for students in Grades 11-12 with knowledge of calculus and inequalities.

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