Let $f(x) = x^3 - 4x +1$. Expand $ (x + h)^3$.[2] Use the formula \(f'(x) = \lim\limits_{h

Question

Let $f(x) = x^3 - 4x +1$. Expand $ (x + h)^3$.[2] Use the formula $f'(x) = \lim\limits_{h \to 0} \frac{f(x +h) - f(x)}{h}$ to show that the derivative of $f(x)$ is $3x^2 - 4$.[4] The tangent to the curve of $f$ at the point $ \mathrm{P} (1,-2)$ is parallel to the tangent at a point $\mathrm{Q}$. Find the coordinates of $\mathrm{Q}$.[4] The graph of $f$ is decreasing for $p < x< q$. Find the value of $p$ and of $q$.[3] Write down the range of values for the gradient of $f$.[2]

Accepted Answer

(a) $x^3+3x^2h+3xh^2+h^3$ (b) $f'(x)=3x^2-4$ (c) $Q=(-1,4)$ (d) $p=-\dfrac{2\sqrt3}{3}$, $q=\dfrac{2\sqrt3}{3}$ (e) $[-4,\infty)$

Binomial Theorem - Binomial Expansion for Positive Integer Powers

Binomial Theorem - Binomial Expansion for Positive Integer Powers

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