The curves are $$\mathrm{f}(x)=x^3,\qquad \mathrm{g}(x)=-2x^3.$$ Find both gradient functions. Find

Question

The curves are $$\mathrm{f}(x)=x^3,\qquad \mathrm{g}(x)=-2x^3.$$ Find both gradient functions. Find both gradients at $x=1$. Explain how the gradients are related and describe the local direction of each curve at $x=1$.

Accepted Answer

(a) $\mathrm{f}'=3x^2,\ \mathrm{g}'=-6x^2$; (b) Gradients $3,-6$; (c) $\mathrm{g}'=-2\mathrm{f}'$, with $\mathrm{f}$ rising and $\mathrm{g}$ falling at $x=1$.

Differentiation

Differentiation

Solution:

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