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f(x) = x2 − 10x (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) x = (b) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) increasing decreasing (c) Apply the First Derivative Test to identify the relative extrema. (Of an answer does not exist, enter DNE.) relative maximum (x, y) = relative minimum (x, y) =

Answer :

Answer:

Required critical point is (5, -25) which takes local minima.

Step-by-step explanation:

Given function is,

[tex]f(x)=x^2-10x\hfill (1)[/tex]

  • To find the critical point we have to do,

[tex]f'(x)=0\implies 2x-10=0\implies x=5[/tex]

Now substitite this value in (1) we get,

[tex]y=f(x)=(5)^2-(10\times 5)=-25[/tex]

Therefore the critical point is (5,-25).

  • To examine critical value takes maxima or minima we have to consider double derivative of (1) at x=5, that is,

[tex]f''(5)=2>0[/tex]

since double derivative at x=5 is geater than zero so critical point (5,-25) takes local minima.

  • There exist no any local maxima.

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