\(\mathrm{f}^{\prime}(x)0:\ \text{increasing}\qquad \mathrm{f}^{\prime}(x)

Question

$\mathrm{f}^{\prime}(x)>0:\ \text{increasing}\qquad \mathrm{f}^{\prime}(x)<0:\ \text{decreasing}$ Solve the derivative inequality on the function’s domain. Split intervals at excluded values. In the sketch, the function is increasing for $xc$, and decreasing for $a0:\ \text{gradient increasing}\qquad \mathrm{f}^{\prime\prime}(x)<0:\ \text{gradient decreasing}$ Example For $\mathrm f(x)=x^3-3x^2$, $\mathrm f'(x)=3x^2-6x=3x(x-2)$. $\mathrm f'(x)>0$ for $x<0$ or $x>2$, so $\mathrm f$ is increasing there; $\mathrm f'(x)<0$ for $0

Accepted Answer

Increasing where $\frac{\mathrm dy}{\mathrm dx}=\mathrm{f}'(x)>0$. Decreasing where $\frac{\mathrm dy}{\mathrm dx}=\mathrm{f}'(x)c$, and decreasing for $a<x<c$.

Increasing and Decreasing Functions

Increasing and Decreasing Functions

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