Download App

Applications of Derivatives (p-1) — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Applications of Derivatives (p-1)2 Marks
Question
Find the values of $x$ such that $f(x)$ is decreasing function:
$f(x)=2 x^3-15 x^2-84 x-7$
✓

Answer

$
\begin{aligned}
& f(x)=2 x^3-15 x^2-84 x-7 \\
& \therefore f^{\prime}(x)=\frac{d}{d x}\left(2 x^3-15 x^2-84 x-7\right) \\
& =2 \times 3 x^2-15 \times 2 x-84 \times 1-0 \\
& =6 x^2-30 x-84 \\
& =6\left(x^2-5 x-14\right)
\end{aligned}
$
$f$ is decreasing, if $f^{\prime}(x)<0$
i.e. if $6\left(x^2-5 x-14\right)<0$
i.e. if $x^2-5 x-14<0$
i.e. if $x^2-5 x<14$
i.e. if $x-5 x+\frac{25}{4}<14+\frac{25}{4}$
i.e. if $\left(x-\frac{5}{2}\right)^2<\frac{81}{4}$
i.e. if $-\frac{9}{2}<x-\frac{5}{2}<\frac{9}{2}$
i.e. if $-\frac{9}{2}+\frac{5}{2}<x-\frac{5}{2}+\frac{5}{2}<\frac{9}{2}+\frac{5}{2}$
i.e. if $-2<x<7$
$\therefore f$ is decreasing, if $-2< x <7$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(2 Marks)" from Applications of Derivatives (p-1)→Applications of Derivatives (p-1) — all question types→Maths (commerce) STD 12 Commerce / Arts question papers→Maharashtra Board STD 12 Commerce / Arts question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following definite integrals : $\int_0^1 \frac{x^2+3 x+2}{\sqrt{x}} d x$
→
Verify that $y=\log x+c$ is a solution of the differential equation
$x \frac{d^2 y}{d x^2}+\frac{d y}{d x}=0$
→
A merchant gives 5% commission and 1.5% delcredere to his agents. If the agent sells goods worth $\text{₹}$ 30,600 how much does he get? How much does the merchant receive?
→
Solve the following differential equations : $\frac{d y}{d x}+ y =3$
→
Examine whether the statement pattern

[p → (~ q ˅ r)] ↔ ~[p → (q → r)] is a tautology, contradiction or contingency.

→
Evalute : $\int e^x \frac{1+x}{(2+x)^2} d x$
→
Find the rate of change of demand $(x)$ of a commodity with respect to its price $( y )$ if $y =25+30 x - x ^2$.
→
Evaluate the following definite integrals : If $\int_0^a(2 x+1) d x=2$, find the real values of ' $a$ '.
→
A stock worth ₹ 7,00,000 was insured for ₹ 4,50,000. Fire burnt stock worth ₹ 3,00,000 completely and damaged there remaining stock to the extent of 75% of its value. What amount can be claimed undertaken policy?
→
Evalute : $\int \frac{e^x}{4 e^{2 x}-1} d x$
→