Given that $y={{x}^{4}}-4x$ Show that $y$ is decreasing for $x

Question

Given that $y={{x}^{4}}-4x$ Show that $y$ is decreasing for $x<1$. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$, $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}$ and ${{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}$, respectively.

Accepted Answer

(i) $y'<0$ for $x<1$, so $y$ is decreasing (shown). (ii) $4x^3-4$, $12x^2$, and $16(x^3-1)^2$, respectively.

Techniques of Differentiation - Second and Higher Derivatives

Techniques of Differentiation - Second and Higher Derivatives

Solution:

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