Question:
For what value of \(x\), the function \(2x^3+3x^2-36x+10\) has minimum?
For what value of \(x\), the function \(2x^3+3x^2-36x+10\) has minimum?
Show Hint
For maximum or minimum, first solve \(f'(x)=0\). Then use \(f''(x)>0\) for minimum and \(f''(x)<0\) for maximum.
Updated On: May 7, 2026
- \(-2\)
- \(-3\)
- \(2\)
- \(1\)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Let the function be
\[
f(x)=2x^3+3x^2-36x+10.
\]
To find the point where the function has a minimum, first find the critical points by differentiating.
\[
f'(x)=\frac{d}{dx}(2x^3+3x^2-36x+10).
\]
\[
f'(x)=6x^2+6x-36.
\]
Take \(6\) common:
\[
f'(x)=6(x^2+x-6).
\]
Factorize:
\[
x^2+x-6=(x+3)(x-2).
\]
So,
\[
f'(x)=6(x+3)(x-2).
\]
For critical points:
\[
f'(x)=0.
\]
\[
6(x+3)(x-2)=0.
\]
Therefore,
\[
x=-3
\]
or
\[
x=2.
\]
Now use the second derivative test.
\[
f''(x)=\frac{d}{dx}(6x^2+6x-36).
\]
\[
f''(x)=12x+6.
\]
At \(x=2\):
\[
f''(2)=12(2)+6=24+6=30.
\]
Since
\[
f''(2)>0,
\]
the function has a minimum at
\[
x=2.
\]
Hence, the required value of \(x\) is
\[
2.
\]
Was this answer helpful?
0
0
Top AP ECET Mathematics Questions
- The sum of the four digit even numbers that can be formed with the digits 0,3,5,4 with out repetition is
- A random variable X has the following probability distribution The variance of this random variable is
- For the matrix $A = \begin{pmatrix}3&-3&4\\ 2&-3&4\\ 0&-1&1\end{pmatrix}, A^{-1} = $
- If $A = \begin{bmatrix}k/2&0&0\\ 0&1/3&0\\ 0&0&m/4\end{bmatrix} $ and $A^{-1} = \begin{bmatrix}1/2&0&0\\ 0&1/3&0\\ 0&0&1/4\end{bmatrix}$ then $ k + l + m = $
- Number of ways of forming a committee of $6$ members out of $5$ Indians, $5$ Americans and $5$ Australians such that there will be atleast one member from each country in the committee is
View More Questions
Top AP ECET Maxima and Minima Questions
Top AP ECET Questions
- The sum of the four digit even numbers that can be formed with the digits 0,3,5,4 with out repetition is
- Thermal decomposition of $HCOOH$ is a first order reaction and the rate constant at T(K) is $4.606 \times 10^{-3}\, s^{-1}$. The time required to decompose $90\%$ of initial quantity of $HCOOH$ at $T(K)$ in seconds is
- Three rods of same dimensions have thermal conductivities 3 K, 2 K and K. They are arranged as shown below:What will be the temperature T of the junction?
- To which group and period does the element belong if the electronic configuration of an element in its $-2$ oxidation state is $1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6}$ ?
- A ball of mass 0.6 kg attached to a light inextensible string rotates in a vertical circle of radius 0.75 m such that it has a speed of $ 5\text{ }m{{s}^{-1}} $ when the string is horizontal. Tension in string when it h horizontal on other side is $ (g=10m{{s}^{-}}2) $
View More Questions