Find the minimum and maximum values of the function y=26 t-t^(2) on [14,26] by comparing values at the critical points and endpoints. (Use symbolic notation and fractions where needed. If the function does not have extreme values, enter DNE.)

Question

Find the minimum and maximum values of the function  y=26 t-t^(2) on  [14,26] by comparing values at the critical points and endpoints. (Use symbolic notation and fractions where needed. If the function does not have extreme values, enter DNE.)

Bytelearn · Accepted Answer

Find Critical Points: To find the extreme values of the function y = 26t - t^2 on the interval [14, 26], we first need to find its critical points by taking the derivative and setting it equal to zero.Derivative Calculation: The derivative of y with respect to t is dy/dt = 26 - 2t. We set this equal to zero to find the critical points: 26 - 2t = 0.Evaluate Endpoints: Solving for t gives us t = 26/2, which simplifies to t = 13. However, since 13 is not in the interval [14, 26], it is not a critical point we need to consider for this problem.Maximum Value Comparison: Since there are no critical points within the interval [14, 26], we only need to compare the values of the function at the endpoints of the interval to find the extreme values.Minimum Value Comparison: We evaluate the function at the lower endpoint, t = 14: y(14) = 26(14) - (14)^2 = 364 - 196 = 168.We evaluate the function at the upper endpoint, t = 26: y(26) = 26(26) - (26)^2 = 676 - 676 = 0.Comparing the values at the endpoints, we find that the maximum value of the function on the interval [14, 26] is 168 at t = 14, and the minimum value is 0 at t = 26.

Find the minimum and maximum values of the function $\newline$ $y=26t-t^{2}$ on $[14,2]$ by comparing values at the critical points and endpoints. $\newline$ (Use symbolic notation and fractions where needed. If the function does not have extreme values, enter DNE.)

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Find the minimum and maximum values of the function \newliney=26t−t2y=26t-t^{2}y=26t−t2 on [14,2][14,2][14,2] by comparing values at the critical points and endpoints.\newline(Use symbolic notation and fractions where needed. If the function does not have extreme values, enter DNE.)

Find the minimum and maximum values of the function $\newline$ $y=26t-t^{2}$ on $[14,2]$ by comparing values at the critical points and endpoints. $\newline$ (Use symbolic notation and fractions where needed. If the function does not have extreme values, enter DNE.)