Let $f(x) = x^{3} + 5 x^{2} - 5 x + 2.$ Which of the following is a correct classification of the critical points of $f$?

Question

BlackTom AI · Accepted Answer

f'(x) = 3x^2 + 10x - 5. Set f'(x)=0 to find critical points.

Solve 3x^2 + 10x - 5 = 0 using the quadratic formula:
x = [-10 ± sqrt(10^2 - 4·3·(-5))] / (2·3)
  = [-10 ± sqrt(100 + 60)] / 6
  = [-10 ± sqrt(160)] / 6
  = [-10 ± 4√10] / 6
  = (-5 ± 2√10) / 3.

Thus the two critical points are x1 = (-5 + 2√10)/3 and x2 = (-5 - 2√10)/3.

Use the second derivative to classify: f''(x) = 6x + 10 (polynomial so C^2, second derivative test applies).

Evaluate f'' at x1:
f''(x1) = 6·[(-5 + 2√10)/3] + 10
        = 2·(-5 + 2√10) + 10
        = -10 + 4√10 + 10
        = 4√10, which is positive. A positive second derivative means f is concave up there, so x1 is a local minimum.

Evaluate f'' at x2:
f''(x2) = 6·[(-5 - 2√10)/3] + 10
        = 2·(-5 - 2√10) + 10
        = -10 - 4√10 + 10
        = -4√10, which is negative. A negative second derivative means f is concave down there, so x2 is a local maximum.

Why other classifications would be wrong: swapping the labels (claiming x1 is a maximum and x2 a minimum) contradicts the signs of f'' at those points. Claiming both are minima or both maxima is impossible here because f'' takes opposite signs at the two distinct critical points; and as a cubic, f cannot have two local minima without a local maximum between them.

Conclusion: f has a local minimum at x = -5/3 + (2√10)/3 and a local maximum at x = -5/3 - (2√10)/3, which matches the provided answer.

Let \(f(x) = x^{3} + 5 x^{2} - 5 x + 2.\) Which of the following is a correct classification of the critical points of \(f\)?单项选择题

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