Consider the function \[ f(x)=x^{4}-98 x^{2}+11, \quad-6 \leq x \leq 15 . \] This function has an absolute minimum value equal to and an absolute maximum value equal to

Step 1 ：Consider the function \(f(x)=x^{4}-98 x^{2}+11\), for \(-6 \leq x \leq 15\).

Step 2 ：This function is a quartic polynomial, which means it has at most three local extrema (maxima or minima). These occur where the derivative of the function is zero.

Step 3 ：To find these points, we need to calculate the derivative of the function, set it equal to zero, and solve for x. The derivative of the function is \(f'(x) = 4x^{3}-196x\).

Step 4 ：The critical points, where the derivative is zero, are \(-7, 0, 7\).

Step 5 ：We also need to evaluate the function at the endpoints of the interval, -6 and 15, because the absolute maximum or minimum could occur there. The values of the function at these points are \(-2390, 11, -2390, -2221, 28586\).

Step 6 ：From these values, we can see that the absolute minimum value of the function is \(-2390\) and the absolute maximum value is \(28586\).

Step 7 ：Final Answer: The absolute minimum value of the function is \(\boxed{-2390}\) and the absolute maximum value is \(\boxed{28586}\).