Applications Of Derivatives Question 309
Question: What are the minimum and maximum values of the function $ x^{5}-5x^{4}+5x^{3}-10 $
[DCE 1999]
Options:
A) - 37, - 9
B) 10, 0
C) It has 2 min. and 1 max. values
D) It has 2 max. and 1 min. values
Show Answer
Answer:
Correct Answer: A
Solution:
$ y=x^{5}-5x^{4}+5x^{3}-10 $ \ $ \frac{dy}{dx}=5x^{4}-20x^{3}+15x^{2} $
$ =5x^{2}(x^{2}-4x+3) $
$ =5x^{2}(x-3),(x-1) $
$ \frac{dy}{dx}=0 $ , gives $ x=0,,1,,3 $
Now, $ \frac{d^{2}y}{dx^{2}}=20x^{3}-60x^{2}+30x $ = $ 10x(2x^{2}-6x+3) $ and $ \frac{d^{3}y}{dx^{3}}=10(6x^{2}-12x+3) $
For $ x=0 $ : $ \frac{dy}{dx}=0,,\frac{d^{2}y}{dx^{2}}=0,,\frac{d^{3}y}{dx^{3}}\ne 0 $ \
Neither minimum nor maximum
For $ x=1,,\frac{d^{2}y}{dx^{2}}=-10=negative $ . \ Maximum value $ {y_{max\text{.}}}=-9 $
For $ x=3,,\frac{d^{2}y}{dx^{2}}=90=positive $ \ Minimum value $ {y_{min\text{.}}}=-37 $ .
BETA
