The question says : the gradient of a curve is give by dy/dx = x^2 - 6x. Find the set of values of x for which y is an increasing function of x.
I didn't understand what they are asking me to do could you solve and explain?
You want x^ - 6x to be positive, so
x^2 - 6x > 0
x(x-6) > 0
so the critical values of x are 0 and 6
look at a number < 0 , say x = -1
your statement is -1(-7) > 0 , which is TRUE
look at a number between 0 and 6, say x=4
your statement is 4(-2) > 0 , which is FALSE
look at a number > 6, say x = 10
your statement is 10(4) > 0 , which is TRUE
so x must be less than 0 OR greater than 6
your function increases for
x < 0 OR x > 6
To determine the set of values of x for which y is an increasing function of x, we need to find the values of x where the derivative dy/dx is positive.
Given that dy/dx = x^2 - 6x, we can find the values of x for which dy/dx > 0.
Let's solve the inequality:
x^2 - 6x > 0
To solve this inequality, we can factorize it by finding the roots:
x(x - 6) > 0
Now we have two cases to consider:
1. Case 1: x > 0
If x > 0, both factors in the inequality are positive. Multiplying two positive numbers gives a positive result, so this case satisfies the inequality.
2. Case 2: x - 6 > 0
If x - 6 > 0, we have x > 6. This means that for x > 6, the inequality is satisfied.
Therefore, we have two ranges for x:
1. x > 0
2. x > 6
Taking the intersection of these two ranges, we find that the set of values for x for which y is an increasing function of x is:
x > 6
In other words, y is an increasing function of x when x is greater than 6.
Certainly! In this question, you are asked to find the set of values of x for which y is an increasing function of x.
To determine when y is increasing or decreasing, we can utilize the given gradient of the curve, which is expressed as dy/dx.
In this case, the gradient function is dy/dx = x^2 - 6x.
To find the values of x for which y is increasing, we need to identify when the gradient dy/dx is positive. This indicates that y is increasing as x increases.
So, we need to solve the inequality dy/dx > 0.
Substituting the given gradient function, we have x^2 - 6x > 0.
To solve this quadratic inequality, we first factor out x: x(x - 6) > 0.
Next, we determine the critical points or values of x that make the left side of the inequality equal to zero.
Setting x = 0 and x - 6 = 0, we find two critical points: x = 0 and x = 6.
Now, we can create a sign chart to identify the regions where x(x - 6) is positive.
On the left side of the number line, we can pick a test value less than 0, such as -1. Plugging this value into x(x - 6), we get (-1)(-7) = 7, which is positive.
Between 0 and 6, we can choose a test value, such as 3. Evaluating x(x - 6) at 3, we get (3)(-3) = -9, which is negative.
Finally, on the right side of the number line, let's select a test value greater than 6, like 7. By substituting this value into x(x - 6), we obtain (7)(1) = 7, which is positive.
Now, we can summarize the results from the sign chart.
When x is less than 0 or greater than 6, x(x - 6) is positive.
Therefore, the set of values of x for which y is an increasing function of x is x < 0 or x > 6.