Increasing and decreasing functions. Find the intervals on whi...

Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = (x - 1)²

Accepted Answer

Welcome back, everyone. Find the intervals where the function G of X equals 2 x minus 32 is increasing and decreasing. So let's recall that the function is increasing if its first derivative is positive. So we want to set G of X greater than 0. And then the function is decreasing if its first derivative is negative. Now, let's identify the derivative itself, right? We want to identify G of X. And we can see that it is a composite function we want to differentiate 2 x minus 32. So first of all, let's apply the power rule. We have our exponential function. We bring down the exponent. The original exponent decreases by 1 unit, so we get 2 multiplied by 2X minus 3, and then we have our inner function, so we apply the chain rule and multiply by the derivative of 2 X minus 3. Now the derivative of 2 X minus 3 is 2. And when we combine the terms, we're going to get 4 multiplied by 2 X minus 3. Now let's identify where the function is increasing, so we set the G of X being greater than 0, and this means that 2 x minus 3 is greater than 0. We can ignore the 4 because we can always divide both sides by 4, right? And this will not change the inequality. The inequality sign will still stay the same, and we are going to get 2 x minus 3. Being greater than 0, which means that 2 X is greater than 3, and then X is greater than 3 halves, right? We can divide both sides by 2. And then G of X is negative elsewhere. So we are essentially saying that 2 X minus 3 is negative. So 2 axes less than 3, and axes less than 3 halves. So we can conclude that the function is increasing. On A In the interval. Now let's see where the derivative is positive. Well, from 3 halves up to infinity. And then the function is decreasing when X belongs to the interval. From negative infinity up to 3 halves because X must be less than 3 halves. And those will be our final answers. So let's label them and thank you for watching.

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.f(x) = (x - 1)²

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Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = (x - 1)²