In the chapter “Application of Derivatives,” we learn how derivatives help in solving real-world and exam-oriented problems such as finding tangents with given properties, maxima/minima of functions, optimization in geometry, and estimating function values using linear approximation. These ideas directly appear in CBSE board questions and are frequently used (in both conceptual and computation-heavy forms) in competitive exams like JEE and NEET.
20
Minutes
15
Questions
1 / -0
Marking
Q1. For the curve , the x-coordinates where the tangent is horizontal are:
Q2. A box is formed by cutting equal squares of side (in cm) from each corner of a rectangular sheet and folding up the sides. The value of in that maximizes the volume is:
Q3. Find the coordinates of the point(s) on the curve at which the tangent line passes through the point .
No such point exists
Q4. Assertion (A): If is twice differentiable on and , then is a point of inflection of .
Reason (R): If is twice differentiable on and is a point of inflection of , then .
Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is not the correct explanation of A.
A is true but R is false.
A is false but R is true.
Q5. For the curve , the normal at a point passes through the origin for which value(s) of ?
Only at
At no point
Q6. Use linear approximation of about to estimate the value of .
Q7. Let for . Using derivatives, find the point where attains its maximum and the maximum value.
Q8. A rectangle is inscribed with its base on the x-axis in the right triangle with vertices so that the upper-right corner of the rectangle lies on the hypotenuse. Find the base and height of the rectangle which gives the maximum area.
base , height
base , height
base , height
base , height
Q9. For , consider . Using derivatives, find the point at which attains its minimum and the minimum value.
has no minimum on
Q10. For real , determine the range of for which the cubic equation has three distinct real roots.
Q11. Use linear approximation (differential) at to approximate .
Q12. At which point on the curve is the slope of the tangent maximum?
Q13. Assertion (A): If is continuous on , differentiable on and for all , then is convex on .
Reason (R): A function is convex on if and only if is non-decreasing on .
Both A and R are true and R is a correct explanation of A.
Both A and R are true but R is not a correct explanation of A.
A is true but R is false.
A is false but R is true.
Q14. For which value(s) of does the cubic equation have a repeated root?
Q15. For which real values of does the equation (with ) have exactly two positive real solutions?