At every point on the curve y = x³ - ax² + x - 5, the tangent makes a positive acute angle with the positive x-axis. Find all possible integer values of a.

Question

Accepted Answer

-1. The derivative dy/dx = 3x² - 2ax + 1 must be positive for all x. For a quadratic with positive leading coefficient to be always positive, its discriminant must be negative: 4a² - 12 < 0, giving -sqrt(3) < a < sqrt(3). The integers in this range are -1, 0, and 1.

At every point on the curve y = x³ - ax² + x - 5, the tangent makes a positive acute angle with the positive x-axis. Find all possible integer values of a.

Solution

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