A curve has the equation $y = x^3 - ax^2 + bx + 1$, where $a$ and $b$ are constants. Find

Question

A curve has the equation $y = x^3 - ax^2 + bx + 1$, where $a$ and $b$ are constants. Find, respectively, the relationship between $a$ and $b$ for which $y$ is always increasing, and the curve has two distinct stationary points. In the case where $a = 8$ and $b = 10$, and the rate of decrease of $x$ is $0.05$ units/s when $x = 1$, find the rate of change of $y$ at this instant.

Accepted Answer

(i) $b\ge\dfrac{a^2}{3}$ (ii) $b<\dfrac{a^2}{3}$ (iii) $0.15\text{ unit s}^{-1}$

Differentiation - Connected Rates of Change

Differentiation - Connected Rates of Change

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