determine whether the mean value theorem can be applied to ( f ) on the closed interval (a,b). (select all…

determine whether the mean value theorem can be applied to ( f ) on the closed interval (a,b). (select all that apply.)\n( f(x)=9 x^{3}, quad1,2 )\n( square ) yes, the mean value theorem can be applied.\n( square ) no, ( f ) is not continuous on (a, b).\n( square ) no, ( f ) is not differentiable on ( (a, b) ).\n( square ) none of the above.\nif the mean value theorem can be applied, find all values of ( c ) in the open interval ( (a, b) ) such that ( f^{prime}(c)=\frac{f(b)-f(a)}{b - a} ). (enter your answers as a comma-separated list. if the mean value theorem cannot be applied, enter na.)\n( c= )

determine whether the mean value theorem can be applied to ( f ) on the closed interval (a,b). (select all that apply.)\n( f(x)=9 x^{3}, quad1,2 )\n( square ) yes, the mean value theorem can be applied.\n( square ) no, ( f ) is not continuous on (a, b).\n( square ) no, ( f ) is not differentiable on ( (a, b) ).\n( square ) none of the above.\nif the mean value theorem can be applied, find all values of ( c ) in the open interval ( (a, b) ) such that ( f^{prime}(c)=\frac{f(b)-f(a)}{b - a} ). (enter your answers as a comma-separated list. if the mean value theorem cannot be applied, enter na.)\n( c= )

Answer

Explanation:

Step1: Check continuity and differentiability

A polynomial function (f(x)=9x^{3}) is continuous and differentiable for all real numbers. So on the interval ([1,2]), (f(x)) is continuous on ([1,2]) and differentiable on ((1,2)). Thus, the Mean Value Theorem can be applied.

Step2: Calculate (f(a)) and (f(b))

Given (a = 1), (b=2), (f(a)=f(1)=9\times1^{3}=9), (f(b)=f(2)=9\times2^{3}=9\times8 = 72)

Step3: Calculate (\frac{f(b)-f(a)}{b - a})

(\frac{f(b)-f(a)}{b - a}=\frac{72 - 9}{2-1}=\frac{63}{1}=63)

Step4: Find (f^{\prime}(x)) and solve (f^{\prime}(c))

(f^{\prime}(x)=\frac{d}{dx}(9x^{3})=27x^{2}). Set (f^{\prime}(c)=27c^{2}) equal to (\frac{f(b)-f(a)}{b - a}=63). So (27c^{2}=63), (c^{2}=\frac{63}{27}=\frac{7}{3}), (c=\pm\sqrt{\frac{7}{3}}). But since (c\in(1,2)), (c = \sqrt{\frac{7}{3}}\approx1.53)

Answer:

Yes, the Mean Value Theorem can be applied. (c=\sqrt{\frac{7}{3}})