Using the Coulomb gas method and standard methods of statistical physics, we compute analytically the joint cumulative probability distribution of the extreme eigenvalues of the Jacobi-MANOVA ensemble of random matrices in the limit of large matrices. This allows us to derive the rate functions for the large fluctuations to the left and the right of the expected values of the smallest and largest eigenvalues analytically. Our findings are compared with some available known exact results as well as with numerical simulations finding good agreement.