PrepWiser › Learn › Mathematics › Applications of DerivativesApplications of Derivatives — JEE Main PYQs & Practice Questions Applications of Derivatives is a Mathematics chapter in the JEE Main syllabus. This page brings together 153 previous-year and practice questions on Applications of Derivatives, organised into 6 topics: Tangents and Normals, Rate of Change, Increasing and Decreasing Functions, Maxima and Minima, Optimization Problems and Rolle's and Mean Value Theorems. Most questions come with a step-by-step solution you can work through free in PrepWiser.
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Tangents and Normals Let the normal at a point P on the curve y² – 3x² + y + 10 = 0 intersect the y-axis at (… If the curves x = y⁴ and xy = k cut at right angles, then (4k)⁶ is equal to __________. The equation of the normal to the curve y = (1+x)²ʸ + cos ²(sin –1 x) at x = 0 is : The number of points on the curve y=54 x⁵-135 x⁴-70 x³+180 x²+210 x at which the normal… If the tangent to the curve y=x³-x²+x at the point (a, b) is also tangent to the curve y… If the tangent to the curve, y = eˣ at a point (c, eᶜ) and the normal to the parabola… If the curve y = ax² + bx + c, x∈R, passes through the point (1, 2) and the tangent line… Let a curve y=f(x), x ∈(0, ∞) pass through the points P(1, 3/2) and Q(a, 1/2). If the… If the tangent at a point P on the parabola y²=3x is parallel to the line x+2y=1 and the… If the tangent to the curve, y = f (x) = xlogₑ x, (x > 0) at a point (c, f(c)) is… Let P(h, k) be a point on the curve y = x² + 7x + 2, nearest to the line, y = 3x – 3… Which of the following points lies on the tangent to the curve x⁴eʸ + 2√(y + 1) = 3 at… If the equation of the normal to the curve y = (x - a)/(x + b)(x - 2) at the point (1… Let S be the set of all the natural numbers, for which the line x/a + y/b = 2 is a… If the normal to the curve y(x) = ∫₀ˣ (2t² - 15t + 10)dt at a point (a, b) is parallel… Let the quadratic curve passing through the point (-1,0) and touching the line y=x at… If the tangent at the point (x₁, y₁) on the curve y = x³ + 3x² + 5 passes through the… If the tangent to the curve y = x³ at the point P(t, t³) meets the curve again at Q… Let l be a line which is normal to the curve y = 2x² + x + 2 at a point P on the curve… If the tangent to the curve y = x + sin y at a point (a, b) is parallel to the line… Rate of Change Increasing and Decreasing Functions Let f : R → R be defined as f(x) = { - 4/3x³ + 2x² + 3x,, x > 0; 3xeˣ,, x ≤ 0 . Then f… The function, f(x) = (3x – 7)x 2/3 , x ∈ R, is increasing for all x lying in : For the function f(x)=sin x+3 x-2/π(x²+x), where x ∈[0, π/2], consider the following two… If 5 f(x)+4 f(1/x)=x²-2, ∀ x ≠ 0 and y=9 x² f(x), then y is strictly increasing in : Let the function f(x) = x/3 + 3/x + 3, x ≠ 0 be strictly increasing in (-∞, α₁) ∪ (α₂… Let f be a real valued function, defined on R - -1, 1 and given by f(x) = 3 logₑ | (x -… Consider the function f : R → R defined by f(x) = { ( 2 - sin ( 1/x ) )|x|,x ≠ 0; 0, x =… Let f(x) = 3sin ⁴x + 10sin ³x + 6sin ²x - 3, x ∈ [ - π/6,π/2 ]. Then, f is : The function f(x) = (4x³ - 3x²)/6 - 2 sin x + ( 2x - 1 )cos x : If 'R' is the least value of 'a' such that the function f(x) = x² + ax + 1 is increasing… Let f:R → R be defined as f(x) = { - 55x,, if x < - 5; 2x³ - 3x² - 120x,, if - 5 ≤ x ≤… Let f:ℝ→ ℝ be a twice differentiable function such that f''(x)>0 for all x∈ ℝ and… Let f:R → R and g:R → R be two functions defined by f(x) = log ₑ(x² + 1) - e⁻ˣ + 1 and… The number of real solutions of x⁷ + 5x³ + 3x + 1 = 0 is equal to ____________. Let (2,3) be the largest open interval in which the function f(x)=2 log ₑ(x-2)-x²+a x+1… The interval in which the function f(x)=xˣ, x>0, is strictly increasing is The number of points, where the curve y=x⁵-20 x³+50 x+2 crosses the x-axis, is… Let λ * be the largest value of λ for which the function f_λ (x) = 4λ x³ - 36λ x² + 36x… For the function f(x)=(cos x)-x+1, x ∈ ℝ, between the following two statements (S1)… Let f : (–1, ∞) → R be defined by f(0) = 1 and f(x) = 1/xlog ₑ( 1 + x ), x ≠ 0. Then the… Maxima and Minima Let A be the region enclosed by the parabola y²=2 x and the line x=24. Then the maximum… Let f(x)=∫₀^(x²) (t²-8 t+15)/eᵗ dt, x ∈ R. Then the numbers of local maximum and local… Let (2α,α) be the largest interval in which the function f(t)=(|t+1|)/t²,t<0, is… The sum of all local minimum values of the function f(x)={1-2 x,, x<-1; 1/3(7+2|x|),, -1… Consider the following three statements for the function f:(0,∞)→ ℝ defined by… The sum of all the local minimum values of the twice differentiable function f : R → R… Let 'a' be a real number such that the function f(x) = ax² + 6x - 15, x ∈ R is… If the function f(x)=(1/x)²ˣ ; x>0 attains the maximum value at x=1/e then : Let f(x)=(x+3)²(x-2)³, x ∈[-4,4]. If M and m are the maximum and minimum values of f… The function f(x)=2 x+3(x)^(2/3), x ∈ ℝ, has If the functions f(x)=x³/3+2 b x+(a x²)/2 and g(x)=x³/3+a x+b x², a ≠ 2 b have a common… The set of all real values of λ for which the function f(x) = ( 1 - cos ²x )( λ + sin x… Suppose f(x) is a polynomial of degree four, having critical points at –1, 0, 1. If T =… The maximum slope of the curve y = 1/2x⁴ - 5x³ + 18x² - 19x occurs at the point : The sum of the absolute maximum and minimum values of the function f(x)=|x²-5 x+6|-3 x+2… Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x⁶ is unity and… max(0≤ x≤ π)(16 sin(x/2)cos³(x/2)) is equal to: The sum of absolute maximum and absolute minimum values of the function f(x) = |2x² + 3x… For the function f(x) = 4log ₑ(x - 1) - 2x² + 4x + 5, x > 1, which one of the following… Let f : R → R be a function defined by f(x) = (x - 3) n₁ (x - 5) n₂ , n₁, n₂ ∈ N. Then… Optimization Problems A wire of length 20 m is to be cut into two pieces. A piece of length l₁ is bent to make… A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square… Let C be the circle of minimum area enclosing the ellipse E: x²/a²+y²/b²=1 with… A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into… The minimum value of α for which the equation 4/(sin x) + 1/(1 - sin x) = α has at least… If the point on the curve y² = 6x, nearest to the point ( 3,3/2 ) is (α, β), then 2(α +… Let P and Q be any points on the curves (x-1)²+(y+1)²=1 and y=x², respectively. The… If xy⁴ attains maximum value at the point (x, y) on the line passing through the points… If P is a point on the parabola y = x² + 4 which is closest to the straight line y = 4x… The maximum value of z in the following equation z = 6xy + y², where 3x + 4y ≤ 100 and… The maximum area of a triangle whose one vertex is at (0,0) and the other two vertices… A square piece of tin of side 30 cm is to be made into a box without top by cutting a… Consider the triangles with vertices A(2,1), B(0,0) and C(t, 4), t ∈[0,4]. If the… Let x, y > 0. If x³y² = 2¹⁵, then the least value of 3x + 2y is A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into… Let A(4,-2), B(1,1) and C(9,-3) be the vertices of a triangle ABC . Then the maximum… Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum… The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the… Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that… Rolle's and Mean Value Theorems If Rolle's theorem holds for the function f(x) = x³ - ax² + bx - 4, x ∈ [1,2] with f'(… Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) =… The position of a moving car at time t is given by f(t) = at² + bt + c, t > 0, where a… For all twice differentiable functions f : R → R, with f(0) = f(1) = f'(0) = 0 The value of c in the Lagrange's mean value theorem for the function ƒ(x) = x³ - 4x² +… Let f and g be twice differentiable even functions on (-2, 2) such that f( 1/4 ) = 0, f(… Let f:[0,1] → R be a twice differentiable function in (0,1) such that f(0)=3 and f(1)=5… Let the function, ƒ:[-7, 0]→R be continuous on [-7,0] and differentiable on (-7, 0). If… If c is a point at which Rolle's theorem holds for the function, f(x) = log ₑ( (x² +… Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) ≥ 1… The function f(x) = x³ - 6x² + ax + b is such that f(2) = f(4) = 0. Consider two… Let f: ℝ → ℝ be a thrice differentiable function such that f(0)=0, f(1)=1, f(2)=-1… Let f:[2,4] → ℝ be a differentiable function such that (x log ₑ x) f′(x)+(log ₑ x)… Let ƒ be any function continuous on [a, b] and twice differentiable on (a, b). If for… Let S be the set of all functions ƒ : [0,1] → R, which are continuous on [0,1] and… Let f : (a, b) → R be twice differentiable function such that f(x) = ∫ₐˣ g(t)dt for a… More Mathematics chapters Frequently asked questions How many Applications of Derivatives questions are there for JEE Main? PrepWiser has 153 Applications of Derivatives questions for JEE Main, grouped into 6 topics such as Tangents and Normals, Rate of Change and Increasing and Decreasing Functions.
Are these Applications of Derivatives questions from previous-year JEE Main papers? Yes — most are genuine previous-year questions from past JEE Main papers, and the rest are practice questions written to match the JEE Main pattern and difficulty.
Where can I see the answers and solutions to these Applications of Derivatives questions? Open any Applications of Derivatives question in PrepWiser (it is free) to attempt it; most come with a step-by-step solution.