Derivatives of order up to two — practice questions

Question 1 · difficulty L3 · understanding

Let $f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3)$, $x \in \mathbb{R}$. Then the value of $f'(5)$ is :

• A. $\dfrac{657}{5}$
• B. $\dfrac{117}{5}$
• C. $\dfrac{2}{5}$
• D. $\dfrac{62}{5}$

Question 2 · difficulty L3 · understanding

$$\text { If } y(x)=\left|\begin{array}{ccc} \sin x & \cos x & \sin x+\cos x+1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{array}\right|, x \in \mathbb{R} \text {, then } \frac{d^2 y}{d x^2}+y \text { is equal to }$$

• A. 28
• B. 27
• C. -1
• D. 1

Question 3 · difficulty L3 · understanding

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $(\sin x \cos y)(f(2 x+2 y)-f(2 x-2 y))=(\cos x \sin y)(f(2 x+2 y)+f(2 x-2 y))$, for all $x, y \in \mathbf{R}$. If $f^{\prime}(0)=\frac{1}{2}$, then the value of $24 f^{\prime \prime}\left(\frac{5 \pi}{3}\right)$ is :

• A. 2
• B. 3
• C. $-$3
• D. $-$2

Question 4 · difficulty L3 · understanding

$$\text { If } f(x)=\left\{\begin{array}{ll} x^3 \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0 & , x=0 \end{array}\right. \text {, then }$$

• A. $$f^{\prime \prime}(0)=0$$
• B. $$f^{\prime \prime}(0)=1$$
• C. $$f^{\prime \prime}\left(\frac{2}{\pi}\right)=\frac{24-\pi^2}{2 \pi}$$
• D. $$f^{\prime \prime}\left(\frac{2}{\pi}\right)=\frac{12-\pi^2}{2 \pi}$$

Question 5 · difficulty L3 · understanding

If $$y(\theta)=\frac{2 \cos \theta+\cos 2 \theta}{\cos 3 \theta+4 \cos 2 \theta+5 \cos \theta+2}$$, then at $$\theta=\frac{\pi}{2}, y^{\prime \prime}+y^{\prime}+y$$ is equal to :

• A. $$\frac{1}{2}$$
• B. 1
• C. $$\frac{3}{2}$$
• D. 2

Question 6 · difficulty L3 · understanding

If $$f(x)=\left|\begin{array}{ccc} 2 \cos ^4 x & 2 \sin ^4 x & 3+\sin ^2 2 x \\ 3+2 \cos ^4 x & 2 \sin ^4 x & \sin ^2 2 x \\ 2 \cos ^4 x & 3+2 \sin ^4 x & \sin ^2 2 x \end{array}\right|,$$ then $$\frac{1}{5} f^{\prime}(0)=$$ is equal to :

• A. 2
• B. 1
• C. 0
• D. 6

Question 7 · difficulty L3 · understanding

Let $$f(x) = 2x + {\tan ^{ - 1}}x$$ and $$g(x) = {\log _e}(\sqrt {1 + {x^2}} + x),x \in [0,3]$$. Then

• A. there exists $$\widehat x \in [0,3]$$ such that $$f'(\widehat x) < g'(\widehat x)$$
• B. there exist $$0 < {x_1} < {x_2} < 3$$ such that $$f(x) < g(x),\forall x \in ({x_1},{x_2})$$
• C. $$\min f'(x) = 1 + \max g'(x)$$
• D. $$\max f(x) > \max g(x)$$

Question 8 · difficulty L3 · understanding

Let $$f$$ and $$g$$ be the twice differentiable functions on $$\mathbb{R}$$ such that $$f''(x)=g''(x)+6x$$ $$f'(1)=4g'(1)-3=9$$ $$f(2)=3g(2)=12$$. Then which of the following is NOT true?

• A. $$g(-2)-f(-2)=20$$
• B. There exists $$x_0\in(1,3/2)$$ such that $$f(x_0)=g(x_0)$$
• C. $$|f'(x)-g'(x)| < 6\Rightarrow -1 < x < 1$$
• D. If $$-1 < x < 2$$, then $$|f(x)-g(x)| < 8$$

Question 9 · difficulty L3 · understanding

Let $$y(x) = (1 + x)(1 + {x^2})(1 + {x^4})(1 + {x^8})(1 + {x^{16}})$$. Then $$y' - y''$$ at $$x = - 1$$ is equal to

• A. 496
• B. 976
• C. 464
• D. 944

Question 10 · difficulty L3 · understanding

If $$f(x) = {x^3} - {x^2}f'(1) + xf''(2) - f'''(3),x \in \mathbb{R}$$, then

• A. $$2f(0) - f(1) + f(3) = f(2)$$
• B. $$f(1) + f(2) + f(3) = f(0)$$
• C. $$f(3) - f(2) = f(1)$$
• D. $$3f(1) + f(2) = f(3)$$