Q1
The rate of change of the area of a circle with respect to its radius $r$ is:
📝 SolutionArea $A = \pi r^2$. Rate of change $\frac{dA}{dr} = 2\pi r$.
Q2
A function $f(x)$ is said to be increasing on an interval if for $x_1 < x_2$ in that interval:
📝 SolutionA function is increasing if larger inputs give larger outputs, i.e. $x_1 < x_2 \implies f(x_1) < f(x_2)$.
Q3
A function $f(x)$ is increasing on an interval where:
📝 SolutionIf the first derivative $f'(x) > 0$ throughout an interval, the function is increasing there.
Q4
For $f(x) = x^2 - 4x + 3$, the function is decreasing on:
📝 Solution$f'(x) = 2x - 4$. Setting $f'(x) < 0$ gives $2x - 4 < 0$, i.e. $x < 2$. So the function decreases for $x < 2$.
Q5
At a point of local maximum, $f'(x)$ and $f''(x)$ satisfy:
📝 SolutionBy the second derivative test, a local maximum occurs where $f'(x) = 0$ (critical point) and $f''(x) < 0$ (concave down).
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Class 12 · Maths
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