Let f(x)=x ln(x). What is the absolute minimum value of f ? Choose 1 answer: (A) 0 (B) (1)/(e) (C) -(1)/(e) (D) f has no minimum value

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Let  f(x)=x ln(x). What is the absolute minimum value of  f ? Choose 1 answer: (A) 0 (B)  (1)/(e) (C)  -(1)/(e) (D)  f has no minimum value

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Find Critical Points: To find the absolute minimum value of the function f(x) = x ln(x), we need to find the critical points of the function by taking the derivative and setting it equal to zero.Calculate Derivative: The derivative of f(x) with respect to x is f'(x). Using the product rule for differentiation, we get: f'(x) = d/dx [x ln(x)] = ln(x) + x * (1/x) = ln(x) + 1.Set Derivative Equal to Zero: Now we set the derivative equal to zero to find the critical points: ln(x) + 1 = 0.Solve for x: To solve for x, we subtract 1 from both sides: ln(x) = -1.Exponentiate Equation: To get x by itself, we exponentiate both sides of the equation: e^(ln(x)) = e^(-1).Determine Critical Point Type: Since e^(ln(x)) is just x, we have: x = e^(-1) = 1/e.Substitute x into Function: Now we need to determine if this critical point is a minimum, maximum, or neither. We can do this by using the second derivative test or by analyzing the behavior of the first derivative around the critical point. Since the function ln(x) is increasing for x > 0, and since the critical point x = 1/e is the only critical point, this suggests that f(x) has an absolute minimum at x = 1/e.Calculate Minimum Value: To find the minimum value of f(x), we substitute x = 1/e into the original function: f(1/e) = (1/e) * ln(1/e).Final Answer: We know that ln(1/e) is the same as ln(1) - ln(e), which simplifies to 0 - 1 because ln(1) = 0 and ln(e) = 1. So, f(1/e) = (1/e) * (-1) = -1/e.Therefore, the absolute minimum value of the function f(x) = x ln(x) is -1/e, which corresponds to answer choice (C).

Let $f(x)=x \ln (x)$ . $\newline$ What is the absolute minimum value of $f$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $0$ $\newline$ (B) $\frac{1}{e}$ $\newline$ (C) $-\frac{1}{e}$ $\newline$ (D) $f$ has no minimum value

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Let f(x)=xln⁡(x) f(x)=x \ln (x) f(x)=xln(x).\newlineWhat is the absolute minimum value of f f f ?\newlineChoose 111 answer:\newline(A) 000\newline(B) 1e \frac{1}{e} e1​\newline(C) −1e -\frac{1}{e} −e1​\newline(D) f f f has no minimum value

Let $f(x)=x \ln (x)$ . $\newline$ What is the absolute minimum value of $f$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $0$ $\newline$ (B) $\frac{1}{e}$ $\newline$ (C) $-\frac{1}{e}$ $\newline$ (D) $f$ has no minimum value