Find the intervals on which
𝑓
(
𝑥
)
=
𝑥
5
−
10
𝑥
2
is concave up.

Question

Find the intervals on which 
𝑓
(
𝑥
)
=
𝑥
5
−
10
𝑥
2
 is concave up.

BlackTom AI · Accepted Answer

To determine concavity, I’ll compute the second derivative of f(x) = x^5 − 10x^2. First, find the first derivative: f'(x) = 5x^4 − 20x. Then, compute the second derivative: f''(x) = 20x^3 − 20 = 20(x^3 − 1). Next, factor the cubic part to locate potential inflection points: x^3 − 1 = (x − 1)(x^2 + x + 1). Note that x^2 + x + 1 has discriminant 1 − 4 = −3, so it has no real roots. Therefore the only real critical point for concavity is x = 1. Now assess the sign of f''(x) on intervals determined by x = 1: - For x > 1, say x = 2, f''(2) = 20(8 − 1) = 140 > 0, so f is concave up on (1, ∞). - For x < 1, say x = 0, f''(0) = −20 < 0, so f is concave down on (−∞, 1). Thus, the function is concave up exactly on the interval (1, ∞). If there were any other candidates for inflection, they would come from roots of f''(x), but the only real root here is x = 1, confirming the interval (1, ∞) as the concave-up region.

MATH-112-301-001 Unproctored Midcourse Exam 3 Practice Exam 2

Find the intervals on which 𝑓 ( 𝑥 ) = 𝑥 5 − 10 𝑥 2 is concave up.

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