Passage

Let g(x) =$$\displaystyle \int_{0}^{x}f\left ( t \right )dt,$$ where f is a function
whose graph is show adjacently.
On the basis of above information, answer te following questions.
Mathematics

Set of values of x in [0,7] for which g(x) is negative is 


ANSWER

(5,7)


SOLUTION
$$g(x)$$ starts decreasing from $$x=3$$
$$\displaystyle g\left ( 4 \right )=\int_{0}^{4}f\left ( t \right )dt=\int_{0}^{3}f\left ( t \right )dt+\int_{3}^{4}f\left ( t \right )dt$$
$$\displaystyle =\frac{9}{2}+\int_{3}^{4}\left ( -3t+9 \right )dt=\frac{9}{2}+\left ( 9t+\frac{3t^{2}}{2} \right )^{4}_{3}$$
$$\displaystyle \frac{9}{2}+\left ( 36-24-27+\frac{27}{2} \right )=3$$ Now $$\displaystyle g\left ( x \right )=\int_{0}^{x}f\left ( t \right )dt$$
$$\displaystyle =\int_{0}^{4}f\left ( t \right )dt+\int_{4}^{x}f\left ( t \right )dt$$      $$\displaystyle 0\leq x\leq 6$$
$$\displaystyle =3+\int_{4}^{x}\left ( -3 \right )dt=3-3\left ( x-4 \right )=15-3x $$
$$\displaystyle g\left ( x \right )=0$$
$$\Rightarrow 15-3x=0$$
$$\Rightarrow x=5$$
$$g(x)$$ becomes zero at $$x=5$$
$$g(x)$$ will be negative in $$(5,7)$$
View Full Answer

Its FREE, you're just one step away


Single Correct Medium Published on 17th 09, 2020
Mathematics

Value of x at which g (x) becomes zero, is


ANSWER

5


SOLUTION
g(x) start decreasing from x=3
$$\displaystyle g\left ( 4 \right )=\int_{0}^{4}f\left ( t \right )dt=\int_{0}^{3}f\left ( t \right )dt+\int_{3}^{4}f\left ( t \right )dt$$
$$\displaystyle =\frac{9}{2}+\int_{3}^{4}\left ( -3t+9 \right )dt=\frac{9}{2}+\left ( 9t+\frac{3t^{2}}{2} \right )^{4}_{3}$$
$$\displaystyle \frac{9}{2}+\left ( 36-24-27+\frac{27}{2} \right )=3$$ Now $$\displaystyle g\left ( x \right )=\int_{0}^{x}f\left ( t \right )dt$$
$$\displaystyle =\int_{0}^{4}f\left ( t \right )dt+\int_{4}^{x}f\left ( t \right )dt$$      $$\displaystyle 0\leq x\leq 6$$
$$\displaystyle =3+\int_{4}^{x}\left ( -3 \right )dt=3-3\left ( x-4 \right )=15-3x $$
$$\displaystyle g\left ( x \right )=0$$
$$\Rightarrow 15-3x=0$$
$$\Rightarrow x=5$$ which lies in  $$[0,6]$$
View Full Answer

Its FREE, you're just one step away


Single Correct Hard Published on 17th 09, 2020
Mathematics

Maximum value of g(x) in x $$\displaystyle \epsilon $$ [0,7] is.


ANSWER

9/2


SOLUTION
$$\displaystyle g\left ( x \right )=\int_{0}^{x}f\left ( t \right )dt$$
$$g'(x)=f(x)$$ From the graph it is clear that
$$\displaystyle f\left ( x \right )> 0$$ in $$\displaystyle x\epsilon \left [ 0,3 \right ]$$ and $$\displaystyle f\left ( x \right )< 0$$ in $$\displaystyle x\epsilon \left [ 3,7 \right ]$$
$$g(x)$$ is increasing in $$[0,3]$$ and $$g(x)$$ is decreasing in $$[3,7]$$
maximum value of $$g(x)$$ occurs at $$x=3$$
$$\displaystyle g\left ( 3 \right )=\int_{0}^{3}f\left ( t \right )dt$$
$$\displaystyle =\int_{0}^{1}1\cdot dt+\int_{1}^{2}\left ( 2t-1 \right )dt+\int_{2}^{3}\left ( 3t+9 \right )dt$$
$$\displaystyle =1+\left ( t^{2}-t \right )^{2}_{1}+\left ( 9t-3\frac{t^{2}}{2} \right )^{3}_{2}$$
$$=\displaystyle 1+\left ( 4-2-0 \right )+\left ( 27-\frac{27}{2}-18+6 \right )=\frac{9}{2}$$
View Full Answer

Its FREE, you're just one step away


Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
Solve: $$\displaystyle \int \dfrac{ \,dx}{\sqrt{2ax-x^2} }$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
$$\displaystyle \int \frac{\left ( 1+\log x \right )^{2}}{x}dx.$$
  • A. $$\displaystyle \frac{1}{2}\left ( 1+\log x \right )^{3}$$
  • B. $$\displaystyle \frac{1}{3}\left ( 1-\log x \right )^{3}$$
  • C. $$\displaystyle \left ( 1+\log x^{2} \right )^{3}$$
  • D. $$\displaystyle \frac{1}{3}\left ( 1+\log x \right )^{3}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium
If $$\displaystyle \int f(x)dx =2\{f(x)\}^{3}+c {\it}$$,  and $$f(x) \neq 0$$   then $$f(x)$$ is
  • A. $$\displaystyle \frac{x}{2}$$
  • B. $$\displaystyle x^{3}$$
  • C. $$\displaystyle \frac{1}{\sqrt{x}}$$
  • D. $$\displaystyle \sqrt{\frac{x}{3}}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Evaluate the integral $$\displaystyle\int_{-3}^{3}|x+1|\ dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Hard
Write a value of 
$$\int { \sqrt { 9+{ x }^{ 2 } }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer