Explain why or why not Determine whether the following stateme...

Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The function f(x) = √x has a local maximum on the interval [0,∞).

Accepted Answer

Welcome back, everyone. Evaluate the behavior of the function p of X equals X cubed on the interval from negative infinity to infinity to determine if it has a local minimum. We're given 4 answer choices. A says yes because the function has a turning point at X equals 0. B says no because the function is always increasing. C yes because the function is always decreasing, and D no, because the function is constant. So if we're considering the monotonicity of the function, we have to identify the derivative. So P X corresponds to the derivative of X cubed which can be found using the power rule. So we get 3 x 2d. And we have to understand that x 2 is always greater than or equal to 0 for all the real x values because it's a square of a number, right? So the only critical point that we get is P X equals 0 at x of 0. Now if x is less than 0. T X is positive, right, because we have 3 multiplied by a negative number squared, and that is going to be positive. On the other hand, if X is greater than 0, we still get a positive derivative. So it shows us that the function is increasing for all the X values except from 0 for now, right? Now 0. Itself is a saddle point. Because the derivative doesn't change its sign. It was positive before 0. It is positive after 0, so x equals 0 is a saddle point which essentially means that. From negative infinity up to infinity, the function is increasing since the first derivative is always positive, and indeed we get a cubic function which gets its saddle point at X equals 0 in this case, but it's always increasing because the first derivative is never negative. So essentially what we can conclude is that the correct answer to this problem corresponds to the answer choice B. No, the function does not contain a local minimum because it is always increasing, and this is the case because the first derivative is always positive, except from the settle point where we don't have a point of extrema, right? Specifically, we're not changing the behavior from. An increasing function to a decreasing one, it still keeps increasing. Thank you for watching.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. The function f(x) = √x has a local maximum on the interval [0,∞).

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The function f(x) = √x has a local maximum on the interval [0,∞).