Theory and Examples

In Exerc...

Question

Theory and Examples

In Exercises 51–54,

d. Over what intervals of x-values, if any, does the function y = f(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.)

y = −1/x

Accepted Answer

Welcome back everyone. For the function H of X equals 5 divided by X, identify the intervals of x values where the function is increasing and decreasing. Let's begin by finding the domain of the function H of X is equal to 5 divided by X. Let's recall the division by 0 is undefined, so we want to make sure that X is not equal to 0. So our function is defined for all real numbers except from x equals. And now if we want to consider the monotonicity of this function, we have to differentiate it. So let's identify H of X, which is basically the derivative of 5 divided by X, and we can differentiate it by factoring out 5. And then multiplying 5 by the derivative of 1 divided by x using the reciprocal rule where simply differentiating X to the power of -1 and now applying the. Power rule we get 5 multiplied by -1 multiplied by x raises to the power of -2, right, which gives us -5 x to the power of -2, which is equal to -5 divided by x2. So we have found our derivative. And what we want to do is simply analyze when our derivative is positive, because if our derivative of X is greater than 0, this means that H of X, the function itself, is increasing, right? So let's analyze this first case. One is our derivative positive? Well, essentially what we want to do is check when 5 divided by X2 is greater than 0, right? And we have a ratio of two numbers 5 divided by X2. We have to understand that x2 is always greater than or equal to 0 for any X in general and in the context of this problem, X2d is strictly greater than 0 because the square of a number is nonnegative, and here we cannot take 0, so we're just saying that X2 is always positive, right? And 5 is always positive as well. So we have a ratio of 2 positive numbers and a negative sign in front, right? So essentially, We have no solutions for this inequality. We are dividing 2 positive numbers. The ratio is going to be positive. And then since we have a negative sign in front, it turns that positive number into a negative number. So for every X that we use, this inequality. Is actually less than 0, not greater than 0, so we can tell that. Our function is never increasing. So the interval where HF X is increasing would be simply none. And now for decreasing. In order to check where the function is decreasing, we have to check the second condition. We already indirectly sold for it, right? We had H of X, but now we want to check where it's negative because if the derivative is negative, then our function is decreasing, and we have previously shown that -5 divided by X2 is going to be always negative. For all acts. Not being equal to 0 because we have a ratio of 2 positive numbers and a negative sign in front. So essentially we can say that our function is decreasing. When X belongs to the interval from negative infinity up to 0 and From 0 up to infinity, so all real numbers except from 0 because our function is not defined as 0. Thank you for watching.

Theory and Examples

In Exercises 51–54,

d. Over what intervals of x-values, if any, does the function y = f(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.)

y = −1/x

Key Concepts

Increasing and Decreasing Functions

Derivative and Critical Points

Behavior of Rational Functions

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Theory and ExamplesIn Exercises 51–54,d. Over what intervals of x-values, if any, does the function y = f(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.)y = −1/x