GATE CSE 1998 | Question: 8

Question

1. Find the points of local maxima and minima, if any, of the following function defined in $0\leq x\leq 6$. $$x^3-6x^2+9x+15$$
2. Integrate $$\int_{-\pi}^{\pi} x \cos x dx$$

Comments

This might help in determining the odd and even function 

Answer 1

(a)$f(x)=x^{3}-6x^{2}+9x+15$

so $f'(x)=3x^{2}-12x+9=0 \Rightarrow x=1,3$

Now $f''(x)=6x-12$

$f''(1) < 0,$ so $x = 1$ is point of local maxima, $f''(3) > 0,$ so $x = 3$ is point of local minima.

Also the end points are $0$ and $6$ . $0$ and $6$ are not critical points, as the derivative is not zero there.

At the left endpoint \( a \)

If \( f'(t_0) < 0 \) (so \( f \) is decreasing to the right of \( a \)), then \( a \) is a local maximum.  

If \( f'(t_0) > 0 \) (so \( f \) is increasing to the right of \( a \)), then \( a \) is a local minimum.  

At the right endpoint \( b \):

If \( f'(t_n) < 0 \) (so \( f \) is decreasing as \( b \) is approached from the left), then \( b \) is a local minimum.  

If \( f'(t_n) > 0 \) (so \( f \) is increasing as \( b \) is approached from the left), then \( b \) is a local maximum.

Reference :

https://mathinsight.org/local_minima_maxima_refresher

At the left endpoint \(x = 0\)
\[
f'(x) > 0 \text{ for } x \in (0,1) \Rightarrow f \text{ is increasing to the right of } 0
\]
\[
\therefore x = 0 \text{ is a local minimum.}
\]

At the right endpoint \(x = 6\)
\[
f'(x) > 0 \text{ for } x \in (3,6) \Rightarrow f \text{ is increasing as } x \text{ approaches } 6
\]
\[
\therefore x = 6 \text{ is a local maximum.}
\]

\[
\boxed{
x = 0, 3 \text{ are local minima, and } x = 1, 6 \text{ are local maxima.}
}
\]

(b) Since $x \cos x$ is an odd function, by the properties of definite integration, answer is $0$.

Comments

Thanks @Nirmal .This also cleared my doubt of 2012 question of maxima and minima.

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$f””(x)=6x-12$

So If we check the value at 1,3 first

at x=0 double derivative of f(x) = -12 Hence Shouldn’t this be the point of Local Minima?

at x=6 double derivative of f(x) = 24 >0 Hence Shouldn’t this be the point of Local Minima?

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@Anurag01a Yes same doubt brother,

we know that if double derivation or 

f''(x)<0 and (Slope)f'(x)=0 means function is Concave down , so the point is Local Maxima

f''(x)>0 and (Slope)f'(x)=0 means function is Concave Up , so the point is Local Minima

from this answer we got is different why ??

Answer 2

Complete analysis of a)

Comments

0 cannot be local minima , it can be absolute but not local , for local minima/ maxima points we should have that x contained in an open interval on both sides of it , whereas 0 is the left most extreme in this case, similarly 6 cannot be a local maxima but can be an absolute maxima

this was according to ncert but according to thomas and finney extremes can be local maxima/ minima