First Derivative Test for Local Minimum

Q. $S_1:$ If $f\left(x\right)$ is an increasing function with upward concavity, then concavity of $f^{-1}\left(x\right)$ is also upwards.
$S_2:$ If $f\left(x\right)$ is decreasing function with upwards concavity, then concavity of $f^{-1}\left(x\right)$ is also upwards.

1. $S_1$ and $S_2$ both are true
2. $S_1$ is true and $S_2$ is false
3. $S_1$ is false and $S_2$ is true
4. $S_1$ and $S_2$ both are false

Q. Consider the functions $f\left(x\right)=\{\begin{array}{cc}|x|,& x\le -1\\ x^{1/5}& -1<x\le 1\\ (2-x{)}^3,& x>1\end{array}\text{and}g\left(x\right)=\frac{1}{(x+2{)}^3}-2x-\cos x$. Let $p$ be the number of critical points on the graph of $f\left(x\right)\text{and}q$ be the number of solutions of $g\left(x\right)=0$, then which of the following option(s) is/are correct?

1. $q=2$
2. $p=2$
3. $p=3$
4. $q=1$

Q.

A differentiable function f(x) will have a local minimum at x = b if -

1. f’(b) = 0 , f’(b-h) < 0 & f’(b+h) > 0

2. f’(b) = 0 , f’(b-h) > 0 & f’(b+h) < 0

3. f’(b) = 0

4. None of the above

Q. Let S be the set of real values of parameter $\lambda$ for which the equation $f\left(x\right)=2x^3-3(2+\lambda )x^2+12\lambda x$ has exactly one local maximum and exactly one local minimum. Then S is a subset of

1. None of these
2. (-4, ∞)
3. (-3, 3)
4. (3, ∞)

Q.

Match the entries of col. I with those of col. II.
$\begin{array}{cc}Column-I& Column-II\\ \left(a\right)f\left(x\right)=\frac{1-x+x^2}{1+x-x^2}\text{on}[0,1]& \left(p\right)Greatestvalueoff=1\\ \left(b\right)f\left(x\right)=2tanx-ta{n}^{2}x\text{on}[0,\frac{\pi }{2}]& \left(q\right)Leastvalueoff=\frac{3}{5}\\ \left(c\right)f\left(x\right)=\frac{2}{\pi }(sin2x-x)\text{on}[-\frac{\pi }{2},\frac{\pi }{2}]& \left(r\right)Leastvalueoff=-1\\ \left(d\right)f\left(x\right)=\frac{1}{2},(x^3-3x^2+6x-2)\text{on}(-1,1)& \left(s\right)Leastvalueoff=-6\end{array}$

1. $\left(a\right)\to \left(p\right),\left(b\right)\to \left(q\right),\left(c\right)\to \left(r\right),\left(d\right)\to \left(s\right)$

2. $\left(a\right)\to (p,q),\left(b\right)\to \left(p\right),\left(c\right)\to (p,r),\left(d\right)\to (p,s)$

3. $\left(a\right)\to \left(q\right),\left(b\right)\to \left(p\right),\left(c\right)\to \left(p\right),\left(d\right)\to \left(s\right)$

4. $\left(a\right)\to \left(s\right),\left(b\right)\to \left(q\right),\left(c\right)\to (p,r),\left(d\right)\to (r,s)$

Q. Let $P\left(x\right)=a_0+a_1{x}^2+a_2{x}^4+a_3{x}^6+...+a_n{x}^{2n}$ be a polynomial in a real variable $x$ with $0<a_0<a_1<a_2...<a_n$. The function $P\left(x\right)$ has

1. neither a maxima nor a minima
2. only one maxima
3. both maxima and minima
4. only one minima

Q. Let $g\left(x\right)=2f\left(\frac{x}{2}\right)+f(2-x)$ and $f^{\prime \prime }\left(x\right)<0,\forall x\in (0,2).$ Then which of the following is (are) TRUE?

1. $g\left(x\right)$ is decreasing in $(0,\frac{4}{3})$
2. $g\left(x\right)$ is decreasing in $(\frac{4}{3},2)$
3. $g\left(\frac{1}{10}\right)<g\left(\frac{11}{10}\right)$
4. $g\left(x\right)$ has a local minimum at $x=\frac{4}{3}$

Q. Find the value of ${}^{\prime }c{}^{\prime }$ in a Rolle's theorem for the function $f\left(x\right)=x^3-3x$ in $[-\sqrt{3},0]$.

Q. Verify Rolles Theorem for the function $f\left(x\right)=x^2+2x-8,x\in [-4,2].$

Q. Let f and g be real valued functions such that f(x+y)+f(x-y)=2f(x).g(y)$\forall x,yϵR$. Prove that, if f(x) is not identically zero and $\left|f\left(x\right)\right|\le 1\forall xϵR$, then $\left|g\left(y\right)\right|\le 1\forall yϵR$.

Q. Let $f:R\to R$ be given by$f\left(x\right)=(x-1)(x-2)(x-5).$ Define $F\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt,x>0.$
Then which of the following options is/are correct?

1. $F$ has two local maxima and one local minimum in $(0,\infty )$
2. $F\left(x\right)\ne 0$ for all $x\in (0,5)$
3. $F$ has a local maximum at $x=2$
4. $F$ has a local minimum at $x=1$

Q. Let $P\left(x\right)=a_0+a_1{x}^2+a_2{x}^4+...+a_n{x}^{2n}$ be a polynomial in a real variable x with $0<a_1<a_2...<a_n$ then P(x) has

1. Neither a maximum nor a minimum
2. Only one absolute minima
3. Only one absolute maxima & only one absolute minima.
4. Only one absolute maxima

Q. Let S be the set of real values of parameter $\lambda$ for which the equation $f\left(x\right)=2x^3-3(2+\lambda )x^2+12\lambda x$ has exactly one local maximum and exactly one local minimum. Then S is a subset of

1. $(-4,\infty )$
2. $(-3,3)$
3. $(3,\infty )$
4. R

Q. If $\mid$ x - 1$\mid$ + $\mid$ x + 1$\mid$ = 2, then find x.

Q. Let $g\left(x\right)=2f\left(\frac{x}{2}\right)+f(2-x)$ and $f^{\prime \prime }\left(x\right)<0,\forall x\in (0,2).$ Then which of the following is (are) TRUE?

1. $g\left(x\right)$ is decreasing in $(0,\frac{4}{3})$
2. $g\left(x\right)$ is decreasing in $(\frac{4}{3},2)$
3. $g\left(\frac{1}{10}\right)<g\left(\frac{11}{10}\right)$
4. $g\left(x\right)$ has a local minimum at $x=\frac{4}{3}$

Q. If $f\left(x\right)=1+2x^2+4x^4+6x^6+....+100x^{100}$ is a polynomial in a real variable $x$, then $f\left(x\right)$ has

1. only one maximum
2. neither a maximum nor a minimum
3. only one minimum
4. none

Q. Assertion :Let $f\left(x\right)=x^2+7x+4$ be a polynomial function, then $f^{\prime }\left(2\right)=11$. Reason: A polynomial function is differentiable everywhere.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q. If $f\left(x\right)=x^3+4x^2+\lambda x+1$ is monotonically decreasing function of $x$ in the largest possible interval $(-2,-\frac{2}{3})$ then $\lambda =$

1. $-1$
2. Has no real value.
3. $2$
4. $4$

Q. Let $f:R\to R$ be given by$f\left(x\right)=(x-1)(x-2)(x-5).$ Define $F\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt,x>0.$
Then which of the following options is/are correct?

1. $F$ has a local minimum at $x=1$
2. $F$ has a local maximum at $x=2$
3. $F$ has two local maxima and one local minimum in $(0,\infty )$
4. $F\left(x\right)\ne 0$ for all $x\in (0,5)$

Q. Let S be the set of real values of parameter $\lambda$ for which the equation $f\left(x\right)=2x^3-3(2+\lambda )x^2+12\lambda x$ has exactly one local maximum and exactly one local minimum. Then S is a subset of

1. None of these
2. (3, ∞)
3. (-3, 3)
4. (-4, ∞)

Q.

A differentiable function f(x) will have a local minimum at x = b if -

1. f’(b) = 0 , f’(b-h) < 0 & f’(b+h) > 0

2. f’(b) = 0 , f’(b-h) > 0 & f’(b+h) < 0

3. f’(b) = 0

4. None of the above

Q.

Match the entries of col. I with those of col. II.
$\begin{array}{cc}Column-I& Column-II\\ \left(a\right)f\left(x\right)=\frac{1-x+x^2}{1+x-x^2}\text{on}[0,1]& \left(p\right)Greatestvalueoff=1\\ \left(b\right)f\left(x\right)=2tanx-ta{n}^{2}x\text{on}[0,\frac{\pi }{2}]& \left(q\right)Leastvalueoff=\frac{3}{5}\\ \left(c\right)f\left(x\right)=\frac{2}{\pi }(sin2x-x)\text{on}[-\frac{\pi }{2},\frac{\pi }{2}]& \left(r\right)Leastvalueoff=-1\\ \left(d\right)f\left(x\right)=\frac{1}{2},(x^3-3x^2+6x-2)\text{on}(-1,1)& \left(s\right)Leastvalueoff=-6\end{array}$

1. $\left(a\right)\to (p,q),\left(b\right)\to \left(p\right),\left(c\right)\to (p,r),\left(d\right)\to (p,s)$

2. $\left(a\right)\to \left(q\right),\left(b\right)\to \left(p\right),\left(c\right)\to \left(p\right),\left(d\right)\to \left(s\right)$

3. $\left(a\right)\to \left(s\right),\left(b\right)\to \left(q\right),\left(c\right)\to (p,r),\left(d\right)\to (r,s)$

4. $\left(a\right)\to \left(p\right),\left(b\right)\to \left(q\right),\left(c\right)\to \left(r\right),\left(d\right)\to \left(s\right)$

Q.

A differentiable function f(x) will have a local minimum at x = b if -

1. f’(b) = 0 , f’(b-h) < 0 & f’(b+h) > 0

2. f’(b) = 0 , f’(b-h) > 0 & f’(b+h) < 0

3. f’(b) = 0

4. None of the above