answer:If g is a constant function, then for any y_1, y_2 in [c,d], we have |g(y_1)-g(y_2)|=0. Therefore, the total variation of g would be 0. Since the total variation of g is positive, it follows that g cannot be a constant function.

answer:The expression can be rewritten by completing the square as follows: -9x^2 + 5x + 10 = -9left(x^2 - frac{5}{9}xright) + 10 To complete the square, we need to add and subtract the square of half the coefficient of x inside the parentheses: -9left(x^2 - frac{5}{9}x + left(frac{5}{18}right)^2 - left(frac{5}{18}right)^2right) + 10 Simplify and group the terms: -9left[left(x - frac{5}{18}right)^2 - frac{25}{324}right] + 10 Now, distribute the -9 and combine the constants: -9left(x - frac{5}{18}right)^2 + frac{225}{36} - frac{25}{36} + 10 Combine the fractions: -9left(x - frac{5}{18}right)^2 + frac{200}{36} + 10 Finally, simplify the fractions: -9left(x - frac{5}{18}right)^2 + frac{50}{9} + frac{90}{9} Combine the fractions again: -9left(x - frac{5}{18}right)^2 + frac{140}{9} This can be written as a mixed number: frac{140}{9} - 9left(x - frac{5}{18}right)^2 = frac{154}{9} - 9left(x - frac{5}{18}right)^2 So the completed square form of the quadratic is: frac{154}{9} - 9left(x - frac{5}{18}right)^2

answer:The ell_2 norm of a vector is calculated by taking the square root of the sum of the squares of its elements. For the given vector left(frac{-3}{2}, frac{-61}{8}, frac{-67}{8}, frac{-13}{8}right), the calculation is as follows: [ ell_2 text{ norm} = sqrt{left(frac{-3}{2}right)^2 + left(frac{-61}{8}right)^2 + left(frac{-67}{8}right)^2 + left(frac{-13}{8}right)^2} ] [ = sqrt{frac{9}{4} + frac{3721}{64} + frac{4489}{64} + frac{169}{64}} ] [ = sqrt{frac{27 + 3721 + 4489 + 169}{64}} ] [ = sqrt{frac{8416}{64}} ] [ = sqrt{131.5} ] [ approx frac{3 sqrt{947}}{8} ] So, the ell_2 norm of the vector is approximately frac{3 sqrt{947}}{8}.

answer:The expression evaluates as follows: [ sum_{k=0}^3 frac{87}{34}sqrt{2}^k = frac{87}{34} left(sqrt{2}^0 + sqrt{2}^1 + sqrt{2}^2 + sqrt{2}^3right) ] [ = frac{87}{34} left(1 + sqrt{2} + 2 + 2sqrt{2}right) ] [ = frac{87}{34} left(3 + 3sqrt{2}right) ] [ = frac{261}{34} + frac{261}{34}sqrt{2} ] [ = frac{261}{34} + frac{87}{17 sqrt{2}} + frac{87}{17} cdot sqrt{2} ] Thus, the simplified form of the sum is: [ frac{261}{34} + frac{87}{17 sqrt{2}} + frac{87 sqrt{2}}{17} ]