Let f:R→R be a function defined by f(x)=∣∣x+2∣−2∣x∥. If m is the number of points of local minima and n is the number of points of local maxima of f, then m+n is
Solutions · 1
Official solution
\[\begin{aligned}
& f(x)=\left\{\[\begin{array}{|cccc|}|-x-2+2 x| & x \leq-2 \\ |x+2+2 x| & -2 \leq x \leq 0 \\ |x+2-2 x| & x \geq 0\end{array}\end{aligned}\]\right. \\
\& f(x)=\left\{\[∣−x−2+2x∣∣x+2+2x∣∣x+2−2x∣x≤−2−2≤x≤0x≥0\]\right. \\
\& f(x)=\left\{\[∣x−2∣∣3x+2∣∣2−x∣x≤−2−2<x≤0x>0\]\right.
\]
\[\begin{gathered}
f(x)=\left\{\[2−x−3x−23x+2x≤−2−2<x≤−32−32<x≤0\end{gathered}\]\right. \\
f(x)=\left\{\[2−x−3x−23x+22−xx−2x≤−2−2<x≤−32−32<x≤00<x<2x≥2\]\right.
\]
\[ No. of maxima =1 No. of minima =2m=2n=1m+n=3\]