Question
On the interval [0, 1] the function $${x^{25}}{\left( {1 - x} \right)^{75}}$$ takes its maximum value at the point
A.
0
B.
$$\frac{1}{4}$$
C.
$$\frac{1}{2}$$
D.
$$\frac{1}{3}$$
Answer :
$$\frac{1}{4}$$
Solution :
$$\eqalign{
& {\text{Let}}\,y = {x^{25}}{\left( {1 - x} \right)^{75}} \Rightarrow \frac{{dy}}{{dx}} = 25{x^{24}}{\left( {1 - x} \right)^{75}} - 75{x^{25}}{\left( {1 - x} \right)^{74}} \cr
& = 25{x^{24}}{\left( {1 - x} \right)^{74}}\left( {1 - x - 3x} \right) = 25{x^{24}}{\left( {1 - x} \right)^{74}}\left( {1 - 4x} \right) \cr
& {\text{For}}\,{\text{maximum}}\,{\text{value}}\,{\text{of}}\,y,\frac{{dy}}{{dx}} = 0 \cr
& \Rightarrow \quad x = 0,1,\frac{1}{4},x = \frac{1}{4} \in \left( {0,1} \right) \cr
& {\text{Also}}\,{\text{at}}\,x = 0,y = 0,\,{\text{at}}\,x = 1,y = 0,\,{\text{and}}\,{\text{at}}\,x = \frac{1}{4},y > 0 \cr
& \therefore {\text{Max}}{\text{.}}\,{\text{value}}\,{\text{of}}\,y\,{\text{occurs}}\,{\text{at}}\,x = \frac{1}{4} \cr} $$