Increasing & decreasing practice · Easy

Option D (-2) is correct because it is the least value of k for which the derivative of the function is greater than 0 in the given interval, ensuring the function is increasing.

Option D (${\pi ^2}$) is correct because it is the result of evaluating the definite integral using properties of definite integrals.

Option D (72) is correct because after finding the largest value of $\lambda$ for which the function is increasing, substituting $\lambda^*$ into the function and evaluating at x=1 and x=-1 gives a sum of 72.

Option A (Increasing on $(-\infty,-1] \cup[1, \infty)$ and decreasing on $(-1,1)$) is correct because the derivative is positive on $(-\infty,-1] \cup[1, \infty)$ and negative on $(-1,1)$.

Option A ($\pm 12$) is correct because for the function to be increasing, its derivative must be greater than 0, leading to the condition that a lies between -12 and 12.

Option B ($\left(\frac{2}{5}, \frac{1}{2}\right) \cup \left(\frac{3}{5}, \frac{4}{5}\right]$) is correct because it represents the range of the function considering the greatest integer function and the given interval.

The correct answer is D (None of these) because the condition for $f(x)$ to be decreasing leads to $b \in(-\sqrt{3}, \sqrt{3})$, which is not covered by the other options.

Option C ($(0,1 / e)$) is correct because the function $f(x)=x^x$ decreases when its derivative is negative, which occurs on the interval $(0,1 / e)$.

Option C (seven roots in $(a,b)$) is correct because $g(x)g'(x) = 0$ has at least 7 roots, derived from the properties of $f(x)$ and its derivatives.

Option A (Increasing in the neighbourhood of $x=a$) is correct because $f'(x) > 0$ in the neighborhood of $x=a$, indicating that $f(x)$ is increasing.

Always 0. Since $h(x) = f(g(x))$, $f$ is increasing, and $g$ is decreasing, $h(x)$ is decreasing. Given $h(0) = 0$ and $h(x)$ maps to $[0, \infty)$, $h(x)$ must always be 0.

Option A (an increasing function) is correct because the derivative of the function is positive for all x in R, indicating an increasing function.

The correct answer is C ((-3 / 4,-1 / 2)) because $f(x)$ has a real root in this interval since $f(-3/4) < 0$ and $f(-1/2) > 0$, and $f(x)$ is strictly increasing.

Option A (-$\frac{{2}}{{\sqrt{{e}}}}$) is correct because it represents the minimum value of the function f(x) found by analyzing its derivatives.

Option B (9) is correct because using the derivative and the given interval, we can determine the value of 'a'.

Option A (2, 2) is correct. The maximum slope occurs where the second derivative of the curve is zero and the third derivative is negative, which happens at x=2. Substituting x=2 into the original equation gives y=2.

The correct answer is A ($f(x)$ is strictly increasing function) because the derivative $f'(x)$ is always positive, indicating that $f(x)$ is strictly increasing.

Option A is correct. The function is increasing where its derivative is positive, which occurs on the intervals $[0, \pi / 4) \cup(5 \pi / 4,2 \pi]$.

Option B ($k > 3$) is correct because for the function to be increasing on R, its derivative must be greater than 0, leading to the condition k > 3.

Option C ($R - \left[ { - 2,\;0} \right]$) is correct because the function is increasing when its derivative is positive, which occurs outside the interval [-2, 0].

Option D ($(-\frac{{\pi}}{{2}}, \frac{{\pi}}{{4}})$) is correct because the function $f(x)=\tan^{-1}(\sin x + \cos x)$ is increasing when its derivative is positive, which occurs on the interval $(-\frac{{\pi}}{{2}}, \frac{{\pi}}{{4}})$.

Option A (${0,\frac{1}{2}} \cup \left[ 1 \right.,\left. \infty \right)$) is correct because it represents the intervals where the derivative of h(x) is greater than or equal to zero, indicating an increasing function.

Option B is correct because the derivative of the function, f'(x), is positive in the interval (0, e), indicating that the function is increasing in that interval.

Option A ($\frac{19}{3} a+1+(\sin 3-\sin 2)$) is correct because it represents the difference between the function's values at the upper and lower limits of the interval [2,3], obtained by integrating the given expression.

Option B is correct because the derivative of the function, f'(x), is negative on the interval [0, ∞), indicating that the function is decreasing.

Option D (Decreases in the interval $\left( {1,\infty } \right)$) is correct because the derivative of the function is negative in that interval, indicating a decreasing function.

The correct answer is D (None of these) because for $0 < x < \frac{{\pi}}{{2}}$, the correct inequality is $\sin x < x < \tan x$.

Option B is correct. The function f(x) is not monotonic on $( - \infty ,\,0) \cup (0,\,\infty )$ because its derivative oscillates.

Option C ($(e, \infty)$) is correct because the function $f(x)=\frac{{x}}{{\log x}}$ increases when its derivative is positive, which occurs on the interval $(e, \infty)$.

Option D is correct because the derivative of the function is non-negative for all real numbers, indicating that the function is always increasing.