Find all critical points of the function f(x)=x^(-14)-x^(-15). (Use symbolic notation and fractions where needed. Give your answer in the form of comma separated list. If the function does not have any critical points, enter DNE.)

Question

Find all critical points of the function  f(x)=x^(-14)-x^(-15). (Use symbolic notation and fractions where needed. Give your answer in the form of comma separated list. If the function does not have any critical points, enter DNE.)

Bytelearn · Accepted Answer

Find Derivative: To find the critical points of the function, we need to find the points where the derivative of the function is either zero or undefined. Let's find the derivative of f(x) = x^(-14) - x^(-15). Using the power rule for derivatives, the derivative of x^n is n*x^(n-1), we get: f'(x) = -14*x^(-15) + 15*x^(-16).Find Critical Points: Now, we need to find the values of x for which f'(x) = 0 or where f'(x) is undefined. Setting f'(x) to zero gives us the equation: -14*x^(-15) + 15*x^(-16) = 0.Solve Derivative Equation: To solve this equation, we can multiply through by x^16 to avoid dealing with negative exponents: -14*x + 15 = 0.Solve for x: Solving for x, we get: x = 15/14.Check for Undefined Derivative: We also need to check where the derivative is undefined. The derivative will be undefined when x = 0, since we cannot divide by zero. However, x = 0 is not in the domain of the original function f(x) = x^(-14) - x^(-15), because we cannot raise 0 to a negative exponent.Final Critical Point: Therefore, the only critical point of the function is x = 15/14.

Find all critical points of the function $\newline$ $f(x)=x^{-14}-x^{-15}.$ $\newline$ (Use symbolic notation and fractions where needed. Give your answer in the form of comma separated list. If the function does not have any critical points, enter DNE.)

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Find all critical points of the function \newlinef(x)=x−14−x−15.f(x)=x^{-14}-x^{-15}.f(x)=x−14−x−15.\newline(Use symbolic notation and fractions where needed. Give your answer in the form of comma separated list. If the function does not have any critical points, enter DNE.)

Find all critical points of the function $\newline$ $f(x)=x^{-14}-x^{-15}.$ $\newline$ (Use symbolic notation and fractions where needed. Give your answer in the form of comma separated list. If the function does not have any critical points, enter DNE.)