Maxima and Minima Example

f(x)= 3x^{4}-4x^{3} +10   has a minimum value at___________

Answer

f(x)= 3x^{4}-4x^{3} +10   

f'(x) = 12x^{3} -12x^{2} 

critical points         :f'(x) = 12x^{3} -12x^{2} =0

                                       12x^{2}(x-1))=0

                                       x=0,1

 f''(x) = 36x^{2} -24x

at x=1 

f''(1) = 36*1^{2} -24*1 =12>0  So at x=1 function has a minimum value.

and the minimum value is -

f(x)=3x^{4} -4x^{2}+10

f(1)=3*1^{4} -4*1^{2}+10=9

at x=0

f''(0) = 36*0^{2} -24*0 =0  then find f'''(x)

f'''(x) =72x-24

f'''(0) =72*0-24=-24 So at x=0 ,f(x) has neither minimum or maximum value.

 

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