Finding the total variation of $3x^2-2x^3$

Question

I would appreciate if someone could help me to find the total variation of $3x^2-2x^3$ on $[-2,2]$. Thanks

Answer 1

Hint:

(1) The given function is monotone ascending in $\;\left[0\,,\,\frac34\right]\;$

(2) The total variation (= t.v.) of a function in $\;[a,b]\;$ equals the sum of its t.v. in $\;[a,c]\,,\,[c,b]\;,\;\;a<c<b$ .

Another, much easier (perhaps) approach: If you already studied it, you can use the theorem that says the t.v. of $\;f\;$ in $\;[a.b]\;$ is

$$\int\limits_a^b|f'(x)|dx$$

as long as $\;f'(x)\;$ exists and is integrable Riemann in that interval.