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As Ron said, the LMV721 (and about 95% of most op-amps) do not like capacitors hung directly on their output - particularly in a "follwer" configuration.
How much current do you need? It is possible to add a series resistor inside the feedback loop to add capacitive isolation but preserve DC balance - but you add some output impedance.
Remember that the op-amps output impedance rises as frequency goes up - so it is not always "zero" - especially at 10MHz. So the capacitors should provide the low impedance down as low a frequency as possible. Use switcher grade tantalums for the large caps and SMT ceramics for the small - not cheap aluminum electrolytics.
We do make high capacitive load drive op-amps, that were originally designed to drive LCD screen VCOM lines or other high capcitence loads. The main difference with these op-amps is their lower output impedance and internal compensation to keep them stable. This is why they have higher supply currents than similar BW amps.
See the LM7321/2 or LM8262 or LM8272 - these are the "unlimited" cap load drive devices.
BTW: I do not like the "unlimited cap load" designation. There is a limit, usually the output current available to charge/discharge the capacitor. If your application is mostly DC with small transients, then the above devices should do fine.