Find the largest interval on which the given function is incre...

Question

Find the largest interval on which the given function is increasing.

c. g(x) = (3x - 1)¹/³

Accepted Answer

Welcome back, everyone. The function F of X equals 4 x + 2 raise to the power of 1/3 is increasing on the largest interval. First of all, let's determine the domain of this function where given F of X equals 4 X + 2 raise the power of 1/3. The reason why we need to identify the domain is simply to exclude any of X values for which our function is not defined, right? So we don't want to consider those X values. Essentially, we have an exponent which is 1/3, and we have to recall that it represents the cube root, right? So we have a cube root of 4 X + 2. And essentially from here we can say that x belongs to all real numbers. Our function is defined for all the real numbers because we're taking an odd root of 4 x + 2, and those can be negative numbers, positive numbers, and 0, right? So X belongs to all real numbers. And now, let's identify where the function is increasing. We're going to depict an error which goes up, saying that our function is increasing. And now we want to specify the condition for this. We want to make sure that our first derivative is positive. So let's identify the first row F using the power rule. We're going to differentiate 4 X. Plus 2, raise the power of 1/3. Let's apply the power rule. This gives us 1/3 multiplied by 4 x + 2. Raise the power of 1/3 minus 1, which is 2/3, and then because we have an inner function based on the chain rule, we are multiplying by the derivative of 4 X + 2. Now let's identify the answer. So first of all, we're going to rewrite the first part as 1 divided by 3, multiplied by 4 x + 2 raise the power of 2/3 using the reciprocal rule, and then we're multiplying by 4 because the derivative of 4 x + 2 is 4, and now we can rewrite it in a radical form as 4. Divided by We Multiplied by cube root. Off 4 x + 2 squared. What can we sell from here? Well, if we carefully analyze our denominator. First of all, we have to notice that there is a square of a number, so we have 4 X + 2 squad. We have to recall that a square of a number. It is never negative in real plane, right? So it's always greater than or equal to 0. In this case, we cannot take a zero value because this would lead to division by zero, right? But basically, what we can say is that this term is greater than 0 for all possible X values, right? If this derivative is greater than 0 for all x values due to the fact that our denominator is always positive, this means that our function is, is increasing for any X, right? So we can conclude that the function is increasing. For acts in the interval from negative infinity to infinity. Let's label our final answer and thank you for watching.

Find the largest interval on which the given function is increasing.

c. g(x) = (3x - 1)¹/³

Key Concepts

Derivative

Critical Points

Increasing and Decreasing Intervals

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