Interpret the function

The given piecewise function is intended as

We are told that $f(x)$ is continuous at $x=0$.

Condition for continuity at $x=0$

For continuity at $x=0$, we need

So compute the limit:

Use the standard limit

Recall:

Therefore,

and

Hence,

Apply continuity

Since $f(0)=8$,

Find $k_1^2+k_2^2$ from the options

The relation obtained is only

Now check the options for possible values of $k_1^2+k_2^2$.

Using

this is not uniquely determined unless there is some further condition on $k_1,k_2$.

However, among standard exam interpretations, $k_1,k_2$ are often taken as positive integers. Then with

possible pairs are:

• $(1,3)$ giving $1^2+3^2=10$
• $(2,2)$ giving $2^2+2^2=8$
• $(3,1)$ giving $10$

So even under integer assumption, the value is still not unique.

Compare with the stored answer

The stored answer is B: 10.

But from the continuity condition alone, we only get

and hence $k_1^2+k_2^2$ cannot be uniquely determined. So the question as written appears incomplete or misprinted.

Conclusion

The mathematically correct conclusion is:

• Continuity implies $k_1+k_2=4$.
• Therefore $k_1^2+k_2^2$ is not uniquely determined from the given information.
• So the stored answer $10$ cannot be justified from the question as stated.