Find all the critical numbers of y equals x to the power of 4 minus 2 x squared minus 3, then determine the local minimum and maximum points by using a graph. Critical numbers: x equals short dash 1, x equals 0, and x equals 1 left parenthesis short dash 1 comma short dash 4 right parenthesis and left parenthesis 1 comma short dash 4 right parenthesis are local minimums and left parenthesis 0 comma short dash 3 right parenthesis is a local maximum. Critical numbers: x equals short dash 1 and x equals 1 left parenthesis short dash 1 comma short dash 4 right parenthesis and left parenthesis 1 comma short dash 4 right parenthesis are local minimums. Critical numbers: x equals 0 and x equals short dash 1 left parenthesis short dash 1 comma short dash 4 right parenthesis is a local and global minimum and left parenthesis 0 comma short dash 3 right parenthesis is a local maximum. Critical numbers: x equals 0 and x equals 1 left parenthesis 1 comma short dash 4 right parenthesis is a local minimum and left parenthesis 0 comma short dash 3 right parenthesis is a local maximum.

The function 𝑓(𝑥,𝑦)=𝑥3−3𝑥−2𝑥𝑦−𝑦2−2𝑦+4 has two critical points. Find and classify them. the left-most stationary point (the one with the lower value of x) is located at x = and y = and is a point. • the right-most stationary point (the one with the higher value of x) is located at x = and y = and is a point

The function 𝑓(𝑥,𝑦)=𝑥3−3𝑥−2𝑥𝑦−𝑦2−2𝑦+4 has two critical points. Find and classify them

To determine where the function \( f(x) = x^2 - 4x + 4 \) is increasing or decreasing, we need to find its first derivative and analyze the critical points. 1. **Find the first derivative of \( f(x) \):** \[ f(x) = x^2 - 4x + 4 \] \[ f'(x) = \frac{d}{dx}(x^2 - 4x + 4) = 2x - 4 \] 2. **Find the critical points:** Critical points occur where the first derivative is zero or undefined. Set \( f'(x) = 0 \): \[ 2x - 4 = 0 \] \[ 2x = 4 \] \[ x = 2 \] 3. **Determine the intervals to test:** The critical point divides the real number line into two intervals: \( (-\infty, 2) \) and \( (2, \infty) \). 4. **Test the sign of \( f'(x) \) in each interval:** - For \( x < 2 \) (e.g., \( x = 0 \)): \[ f'(0) = 2(0) - 4 = -4 \quad (\text{negative}) \] - For \( x > 2 \) (e.g., \( x = 3 \)): \[ f'(3) = 2(3) - 4 = 2 \quad (\text{positive}) \] 5. **Determine where the function is increasing or decreasing:** - \( f'(x) < 0 \) for \( x < 2 \), so \( f(x) \) is decreasing on \( (-\infty, 2) \). - \( f'(x) > 0 \) for \( x > 2 \), so \( f(x) \) is increasing on \( (2, \infty) \). Therefore, the correct answer is: \[ \boxed{E} \]

Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)f(x) = 3x4 + 8x3 − 48x2

For the function of two variables, f (x, y) = x3 − y2 + 9xy + 5. Find the critical points and determine their nature to complete the following sentences (making sure you use at least 3 decimal places in your answer): • the left-most stationary point (the one with the lower value of x) is located at x = and y = and is a point. • the right-most stationary point (the one with the higher value of x) is located at x = and y = and is a point