Classifying Critical Points and Extreme Values for a Function

Key Questions

• How do you find and classify the critical points of #f(x)=x^3#?

  Here is how to find and classify a critical point of #f#.

  Remember that #x=c# is called a critical value of #f# if #f'(c)=0# or #f'(c)# is undefined.

  #f'(x)=3x^2=0 Rightarrow x=0# is a critical number.

  (Note: #f'# is defined everywhere, #0# is the only critical value.)

  Observing that #f'(x)=3x^2 ge 0# for all #x#,

  #f'# does not change sign around the critical value #0#.

  Hence, #f(0)# is neither a local maximum nor a local minimum by First Derivative Test.

  Wataru · · Sep 28 2014

• How do you find and classify the critical points of #f(x)=x^3#?
• How do you find the critical points of a rational function?
• How do you know how many critical points a function has?
• How many critical points can a cubic function have?
• How many critical points can a function have?
• How many critical points can a quadratic polynomial function have?
• What is the first step to finding the critical points of a function?
• How do you find the absolute extreme values of a function on an interval?
• How do you find the extreme values of the function and where they occur?
• What is the extreme value of a quadratic function?
• How do you calculate #f '(x)# and use calculus to find the maximum value of #sin (ln x)# on the interval [1, 10]?
• What is the local maximum and or miminum of the function #f(x)=2x^2+2x^2-4x#?
• How do you use the closed-interval method to find the absolute maximum and minimum values of the function #f(x)=x-2sinx# on the interval #[-π/4, π/2]#?
• How do you find the absolute maximal and minimal values of #f(x)=x^3−6x^2+9x+4# on the interval #[−1, 5]#?
• What is the overall maximum value of the function #f(x)=(18x)/(4+x^2)# on the interval (0,4)?
• How do you find all local maximum and minimum for #g(x)=6 x^3−(144 )x^2 +(1080 )x−4#?
• How do you find the relative maxima and minima of the function #f(x)= 7(x^2 - 16)^2#?
• How do you find the max and min for #f(x)= x - ((64x)/(x+4))# on the interval [0,13]?
• What are the maximum and minimum values that the function #f(x)=x/(1 + x^2)#?
• How do you determine the maximum and minimum value of the function #f(x)=-xe^x + 2#?
• How to find critical points and the highest point on the graph of #4x^2 + 4xy+7y^2=15#?
• How do you find the exact minimum value of #f(x) = e^x + e^(-2x)# on [0,1]?
• How do you use the Extreme Value Theorem to determine the global extrema of the function #f(x)=5+ (x^2)/(x+2)# on the closed interval [-1,3]?
• Question #15c3e
• What are the values and types of the critical points, if any, of #f(x) = abs(x^2-1)#?
• What are the values and types of the critical points, if any, of #f(x) =4x^3+48x^2+26x-246#?
• What are the values and types of the critical points, if any, of #f(x) =x^3 + 3x^2-24x#?
• What are the values and types of the critical points, if any, of #f(x)=300/(1+.03x^2)#?
• What are the values and types of the critical points, if any, of #f(x)=xlnx#?
• What are the values and types of the critical points, if any, of #f(x)=x^2-5x+4#?
• What are the values and types of the critical points, if any, of #f(x)=x / (x^2 + 4)#?
• What are the values and types of the critical points, if any, of #f(x)=x^3-6x^2+12x-6#?
• What are the values and types of the critical points, if any, of #f(x)=cos^2 x - sin^2 x #?
• What are the values and types of the critical points, if any, of #f(x,y)=2x + 4y + 4#?
• What are the values and types of the critical points, if any, of #f(x,y)=y^2-x^2+3x-sqrt(2xy-4x)#?
• What are the values and types of the critical points, if any, of #f(x)=x^3 + 3x^2 + 1#?
• What are the values and types of the critical points, if any, of #f(x)=x^4 - 8x^3 + 18x^2 - 27#?
• What are the values and types of the critical points, if any, of #f(x,z) = x^4 + 15z^2 + 2xz^2 - 456z^2#?
• What are the values and types of the critical points, if any, of #f(x,y) = x(2-x-y)#?
• What are the values and types of the critical points, if any, of #f(x,y) = sqrty(5-2x^2-3y ))#?
• What are the values and types of the critical points, if any, of #f(x)=7x^4-6x^2+1 #?
• What are the values and types of the critical points, if any, of #f(x)=x^4/(6x^2+x)#?
• What are the values and types of the critical points, if any, of #f(x)=3e^(-2x^2))#?
• What are the values and types of the critical points, if any, of #f(x)=e^(-2x^2)-xe^x#?
• What are the values and types of the critical points, if any, of #f(x)=(2x^2+5x+5)/(x+1)#?
• What are the values and types of the critical points, if any, of #f(x) = (x^4/5)(x-4)^2#?
• What are the values and types of the critical points, if any, of #f(x) = (x-5)/(x-3)^2#?
• What are the values and types of the critical points, if any, of #f(x) =x^2-sqrtx#?
• What are the values and types of the critical points, if any, of #f(x,y)=xy(1-8x-7y)#?
• What is the difference between a critical point and a stationary point?
• What are the values and types of the critical points, if any, of #f(x)=(x+3x^2) / (1-x^2)#?
• What are the values and types of the critical points, if any, of #f(x)=2x^5(x-3)^4#?
• What are the values and types of the critical points, if any, of #f(x,y)=4y(1 - x^2)#?
• What are the values and types of the critical points, if any, of #f(x, y) = x^3+y^3-3*x*y-7#?
• What are the values and types of the critical points, if any, of #f(x, y) = xy^2+xy-3x^2y-7x#?
• Is it possible to determine the critical points of a function without using the function's derivatives?
• How do you find the maximum and minimum values of the function #f(x)= x - ((64x)/(x+4))# on the interval [0,13]?
• How do you find the relative extrema for #f(x) = x^2(6-x)^3#?
• How do you find the relative extrema for #f(x) = (x)3^-x#?
• How do you find the relative extrema for #f(x) = x - log_4x#?
• How do you find the relative extrema for #f(x) = (x^2 - 3x - 4)/(x-2)#?
• How do you find the relative extrema for #f(x) = (x+2)/x#?
• How do you find the relative extrema for #f(x)=x^3-4x^2+x+6#?
• How do you find the relative extrema for #f(x) =2x- 3x^(2/3) +2#?
• How do you find the relative extrema for #f(x)=(x^4)+(4x^3)-12#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #2x^3 + 3x^2 - 12x#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x)=x^3-3x^2+1#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x)=x^3+5x^2#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x) = 3x^4 + 16x^3 + 24x^2 + 32#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for # y=cos^2 x - sin^2 x#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for # f (x) = x + 4/x #?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for # f(x) = sqrt(x^2+1) #?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x)=(x^3)/(x^2-4)#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f (x) = x^3 + 6x^2#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x)= 3x^4+4x^3-12x^2+5#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #F(x)= (x^2)(lnx)#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for # f(x) = x² + 2x - 3#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #y = x^3 - 3x^2 - 9x +15#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x) = (x - 1)/x#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x) = x^2e^x - 3#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x) = 2x + ln x#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x)= 2x+3/x#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #y=x^4-2x^3#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x) = x ln x#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for # f(t) = 6t + 1/t#?
• How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for # 3x^5 - 5x^3#?
• How do you find the local max and min for #f(x) = (3x) / (x² - 1)#?
• How do you find the local max and min for #f(x) = 2 x + 3 x ^{ -1 } #?
• How do you find the local max and min for #f(x) = x / (x^2 + 81)#?
• How do you find the local max and min for # y = 3x^4 + 4x^3 – 12x^2 + 1#?
• How do you find the local max and min for # 2x^3-3x^2-36x-3#?
• How do you find the local max and min for #f(x)=-sinx-cosx#?
• How do you find the local max and min for #f(x) = 2x^3 - 5x +1# in the interval is (-3,3)?
• How do you find the local max and min for # f(x)=x^3-2x+5# on the interval (-2,2)?
• How do you find the local max and min for #f (x) = x^(3) - 6x^(2) + 5#?
• How do you find the local max and min for #f(x) = e^x(x^2+2x+1)#?
• How do you find the local max and min for #f(x) = x^4 - 2x^2 + 1#?
• How do you find the local max and min for #y=2x^3 - 5x^2 - 4x + 7#?
• How do you find the local max and min for #x^5 ln x#?
• How do you find the local max and min for #f(x)= 5x - 3#?
• How do you find the local max and min for #f(x)= x^2 + 8x -12#?
• How do you find the local max and min for #f(x)= sinx#?
• How do you find the local max and min for #f(x) = 7x + 9x^(-1)#?
• How do you find the local max and min for #f(x) = x^3 - 27x#?
• How do you find the local max and min for #f(x) = 1 - sqrt(x)#?
• How do you find the local max and min for #f(x)=2x^3 + 5x^2 - 4x - 3#?
• How do you find the local max and min for #x^3-2x^2-8x#?
• How do you find the local max and min for #f(x)= (x^2)/(x-2)^2#?
• How do you find the local max and min for #F(x) = ln((x^4) + 27)#?
• How do you find the local max and min for #f(x)=x+(1/x)#?
• How do you find the local max and min for #y = x^3 − 12x + 9#?
• Question #da337
• What is the minimum value of the function #y=sqrt(x²+2ax+10a²)# where a > 0 ?
• Given #f'(x) = (x+1)(x-2)²g(x)# where #g# is a continuous function and #g(x) < 0# for all #x#. On what interval(s) is #f# decreasing?
• How do you determine the absolute extreme values for the function #y=x(sqrt(1-x²))# on its domain?
• How do you find the absolute and local extreme values for #y=-x+7# on the interval [-10,10]?
• How do you find the absolute and local extreme values for #f(x)=(x-3)^2-9# on the interval [-8,-3]?
• How do you find the critical points for #f(x)=-x^2+6x+2#?
• How do you find the critical points for #f(x)=x^3-2x^2+3x#?
• How do you find the critical points for #y=x^4-3x^3+5#?
• How do you find the critical points for #g(x)=2x^3-3x^2-12x+5#?
• How do you find the critical points for #y=x-sqrtx#?
• How do you find the critical points and local max and min for #y=4x-x^2#?
• How do you find the critical points and local max and min for #y=(x-1)^4#?
• How do you find the critical points and local max and min for #g(x)=2x^3-24x+5#?
• How do you find the critical points to sketch the graph #f(x)=7+6x-x^2#?
• How do you find the critical points to sketch the graph #g(x)=x^4-8x^2-10#?
• How do you find the critical points to sketch the graph #y=x(x+2)^2#?
• How do you find the critical points to sketch the graph #h(x)=27x-x^3#?
• Question #2b61c
• Question #e5c2c
• How do you find the relative extrema of #f(x)=3x^5-5x^3#?
• How do you find the maxima and minima of the function #f(x)=x^3+3x^2-24x+3#?
• Question #be380
• Question #c8903
• Question #cbfb8
• Question #2237b
• Consider the function #y = e^x# defined on #[-1, 2]#. What are the maximum and minimum of the function on this interval?
• Question #a621d