SOLUTION: find the extrema of the ff. function on the given interval, if there are any. determine the values of x at which the extrema occur. f(x)= x the square root of 4-x^2 on [-1,2].

Question 1193571: find the extrema of the ff. function on the given interval, if there are any. determine the values of x at which the extrema occur.
f(x)= x the square root of 4-x^2 on [-1,2].
Found 2 solutions by parmen, ikleyn:
Answer by parmen(42)   (Show Source): You can put this solution on YOUR website!
Sure, I've been improving my skills at solving these simplification problems. Let's find the extrema of the function:
$$f(x)= x\sqrt{4-x^2}$$
On the interval $[-1, 2]$.
We can find the extrema of a function by finding its critical points, which are the points where the derivative is zero or undefined.
**Steps to solve:**
**1. Differentiate the function:**
$$f'(x)=\sqrt{4-x^2}-\frac{x^2}{\sqrt{4-x^2}}$$
**2. Set the derivative equal to zero and solve for x:**
$$f'(x)=0$$
$$\sqrt{4-x^2}-\frac{x^2}{\sqrt{4-x^2}}=0$$
$$\sqrt{4-x^2}=\frac{x^2}{\sqrt{4-x^2}}$$
$$4-x^2=x^2$$
$$2x^2=4$$
$$x^2=2$$
$$x=\pm\sqrt{2}$$
**3. Evaluate the function at the critical points and endpoints of the interval:**
$$f(-1)=-1\sqrt{4-(-1)^2}=-\sqrt{3}$$
$$f(-\sqrt{2})=-\sqrt{2}\sqrt{4-(-\sqrt{2})^2}=-2$$
$$f(\sqrt{2})=\sqrt{2}\sqrt{4-(\sqrt{2})^2}=2$$
$$f(2)=2\sqrt{4-2^2}=0$$
**4. Compare the values of the function at the critical points and endpoints to determine the extrema:**
The maximum value of the function is $2$ at $x=\sqrt{2}$.
The minimum value of the function is $-2$ at $x=-\sqrt{2}$.
**Answer:**
The extrema of the function are:
* Maximum value: $2$ at $x=\sqrt{2}$
* Minimum value: $-2$ at $x=-\sqrt{2}$

Answer by ikleyn(52754)   (Show Source): You can put this solution on YOUR website!
.

        The solution in the post by @parmen is INCORRECT.

It is incorrect, because it uses in the analysis the value = -2,
while the argument is OUT of the domain of the function, which (the domain) is [-1,2].

The correct answers are

The extrema of the function are: * Maximum value: 2 at x = * Minimum value: at x = -1

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