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.

View Explanation

Log in for full answers

Similar Questions

Which one of the following statements is true regarding the function below, on the interval [a,b]?

Which one of the following statements is true regarding the function below, on the interval [a,b]?

Which of these graphs most resembles the graph of at ?

The function 𝑓 ( 𝑥 ) = 1 𝑒 2 𝑥 − 𝑥 2 is concave for

A function has its first derivative graphed below. Based on the graph of above, on which interval(s) is concave down? Select all that apply.

A function 𝑓 has its first derivative graphed below. Based on the graph of 𝑦 = 𝑓 ′ ( 𝑥 ) above, on which interval(s) is 𝑓 concave down? Select all that apply.

The function is concave down on the interval

The function x2−3x+6 is concave.

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler