Finding the Minimum Gradient

A minimum gradient is a different target

If asked for the smallest gradient, minimise $f'(x)$ , not $f(x)$ . Completing the square is often quickest when the derivative is quadratic.

Worked example

Find the minimum gradient of y = x³ − 3x² + 1.

\begin{aligned}y'&=3x^2-6x\\&=3(x-1)^2-3\end{aligned}

The square is non-negative, so the minimum gradient is $-3$ , at $x=1$ . If the point is requested, substitute into the original curve: $(1,-1)$ . This is not a stationary point because its gradient is not zero.

Your turnPractice

For $y=x^3-6x^2+9x+2$ , find the minimum gradient and the point where it occurs.

Show worked answer

$y'=3(x-2)^2-3$ , so the minimum gradient is $-3$ at $x=2$ . The point is $(2,4)$ .

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