A curve has equation $y = \frac{x^{n+1}}{n+1} + \frac{a}{x}$ for some non-zero constants $a$ and

Question

A curve has equation $y = \frac{x^{n+1}}{n+1} + \frac{a}{x}$ for some non-zero constants $a$ and $n$, with $n\ne-1$. The curve has a common gradient value of $\frac{5}{2}$ at both $x = \sqrt{2}$ and $x = \frac{1}{\sqrt{2}}$. By using the substitution $b = 2^{\frac{n}{2}}$, find the equation of the curve.[7] By considering the expansion of $\left( x - \frac{1}{x} \right)^2$, show that the gradient of the curve at all points on the curve is at least $2$.[3]

Accepted Answer

(i) $y=\dfrac{x^3}{3}-\dfrac1x$ (ii) $\dfrac{\mathrm dy}{\mathrm dx}\geq2$

NJC IP4 Techniques of Differentiation Tutorial Q1

NJC IP4 Techniques of Differentiation Tutorial Q1

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