CAT 2021 Set-3 | Quantitative Aptitude | Question: 12

1 vote
59 views

If $f(x) = x^{2} – 7x$ and $g(x) = x + 3,$ then the minimum value of $f(g(x)) – 3x$ is

1. $-16$
2. $-15$
3. $-20$
4. $-12$

retagged

1 vote

Given that,

• $f(x) = x^{2}-7x$
• $g(x) = x+3$

Now, $f(g(x))-3x = f(x+3)-3x = (x+3)^{2} – 7(x+3)-3x$

$\qquad \qquad \qquad = x^{2}+9+6x-7x-21-3x = x^{2}-4x-12$

Let $h(x) = x^{2}-4x-12$

$\Rightarrow h’(x) = 2x-4$

For minimum value of $x:$

$\Rightarrow h’(x) = 0.$

$\Rightarrow 2x-4=0$

$\Rightarrow \boxed{x=2}$

Now, $h(2) = 2^{2}-4(2)-12$

$\Rightarrow h(2)=4-8-12.$

$\Rightarrow \boxed{h(2) = -16}$

$\therefore$ The minimum possible value of $f(g(x))-3x$ is $-16.$

Correct Answer $:\text{A}$

10.1k points 4 8 30
edited

Related questions

1 vote
1
65 views
Consider a sequence of real numbers $x_{1}, x_{2}, x_{3}, \dots$ such that $x_{n+1} = x_{n} + n – 1$ for all $n \geq 1.$ If $x_{1} = -1$ then $x_{100}$ is equal to $4950$ $4850$ $4849$ $4949$
1 vote
2
130 views
Anil can paint a house in $12 \; \text{days}$ while Barun can paint it in $16 \; \text{days}.$ Anil, Barun, and Chandu undertake to paint the house for $₹ \; 24000$ and the three of them together complete the painting in $6 \; \text{days}.$ If Chandu is paid in proportion to the work done by him, then the amount in $\text{INR}$ received by him is
1 vote
For a real number $a,$ if $\dfrac{\log_{15}a + \log_{32}a}{(\log_{15}a)(\log_{32}a)} = 4$ then $a$ must lie in the range $a>5$ $3<a<4$ $4<a<5$ $2<a<3$
In a triangle $\text{ABC}, \angle \text{BCA} = 50^{\circ}. \text{D}$ and $\text{E}$ are points on $\text{AB}$ and $\text{AC},$ respectively, such that $\text{AD = DE}.$ If $\text{F}$ is a point on $\text{BC}$ such that $\text{BD = DF},$ then $\angle \text{FDE, in degrees},$ is equal to $96$ $72$ $80$ $100$
Bank $\text{A}$ offers $6 \%$ interest rate per annum compounded half yearly. Bank $\text{B}$ and Bank $\text{C}$ offer simple interest but the annual interest rate offered by Bank $\text{C}$ is twice that of Bank $\text{B}.$ ... same amount in Bank $\text{A}$ for one year. The interest accrued, in $\text{INR},$ to Rupa is $3436$ $2436$ $2346$ $1436$