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$\begin{array}{1 1} a \\ b \\ c \\ d \end{array}$

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The whole space is divided into 3 regions

1) $0 < r< R_1 \qquad F(r)=0$

2) $R_2 < r< R_2 \qquad F(r)=\large\frac{4}{3} $$\pi G\rho m \bigg( r - \large\frac{R_1^3}{r^2}\bigg)$

3) $R_2 < r< \infty \qquad F(r)=\large\frac{4}{3} $$\pi G\rho m \bigg( \large\frac{R_2^3-R_1^3}{r}\bigg)$

In region 1 - $F=0$

In region 2 - $F$ increase gradually

In region 3 - $F$ decrease gradually

Hence c is the correct answer.

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