Mathematics

# Let $\displaystyle f(x)=ax^{3}+bx^{2}+cx$ have relative extrema x=1 and at $\displaystyle x=5$.If $\displaystyle \int_{-1}^{1}f(x)dx=6$ then

$a=-1$

$b=9$

##### SOLUTION
$\displaystyle \int _{ -1 }^{ 1 }{ f\left( x \right) } dx=6\Rightarrow \int _{ -1 }^{ 1 }{ \left( { ax }^{ 2 }+{ bx }^{ 2 }+cx \right) } dx=6$
$\displaystyle \Rightarrow \left[ \frac { { ax }^{ 4 } }{ 4 } +\frac { { bx }^{ 3 } }{ 3 } +\frac { { cx }^{ 2 } }{ 2 } \right] _{ -1 }^{ 1 }{ =6 }$
$\displaystyle \Rightarrow \left[ \frac { a }{ 4 } +\frac { b }{ 3 } +\frac { c }{ 2 } -\frac { a }{ 4 } +\frac { b }{ 3 } -\frac { c }{ 2 } \right] =16$
$\displaystyle \Rightarrow \frac { 2b }{ 3 } =6\Rightarrow b=9$  ...(1)
$f\left( x \right) ={ ax }^{ 3 }+{ bx }^{ 2 }+cx\\ f'\left( x \right) ={ 3ax }^{ 2 }+bx+c$
$\therefore f'\left( 1 \right) =0\Rightarrow 3a+bc+c=0$   ...(2)
$f'\left( 5 \right) =0\Rightarrow 15a+5b+c=0$   ...(3)
From (1),(2) and (3)
$a=-1$

Its FREE, you're just one step away

Multiple Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Subjective Medium
$\int \ sinx\;\; d(\ cos x) =$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the given integral.
$\displaystyle \int { \cfrac { \sin { \sqrt { x } } }{ \sqrt { x } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
What is the value of $\int_{0}^{\pi}\dfrac {dx}{5-4\cos x}$?
• A. $dfrac {4\pi}{7}$
• B. $dfrac {\pi}{3}$
• C. $None\ of\ these$
• D. $\dfrac {\pi \log 2}{32}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium

$\displaystyle \int_{0}^{1}\tanh xdx=$
• A. $\log(\mathrm{e}+1/\mathrm{e})$
• B. $\log$ $(e-1/e)$
• C. $\displaystyle \log(\frac{\mathrm{e}}{2}-\frac{1}{\mathrm{e}})$
• D. $\log(\mathrm{e}/2+1/2\mathrm{e})$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
Find $\int\limits_0^{\sqrt 2 } {\sqrt {2 - {x^2}} dx}$
• A. 8
• B.
• C. 2
• D. $\frac{\pi }{2}$